Moving away from a pure element (covered by other answers) to a material, something that has a special property.

Consider something that is made rare by the very presence of your intergalactic society. Let's suppose your warp drives give off low-levels of some kind of radiation, so any alloys destined for sensitive equipment need to be space-smelted in special batches. A real-world example of this kind of thing would be Low Background Steel.

Another commodity that your society presumably still values is time. So any material that needs to be aged would have a very high value. Wood that needs to dry for 20 years. Trees that take 200 years to grow (sure, you can genetically modify them to grow quicker, but it's not the same). The scales of a fish that only sheds them once a millennium.

Why Things Are Valuable

Basic economic theory says that a thing will have a higher cost as it's demand outpaces it's supply. Note that this doesn't actually take rarity into account at all. Rare things are expensive because they have an inherently small supply, but you can also have a relatively common thing be expensive just by increasing the demand for it massively. Think about the early spice trade, and how lucrative that was despite spices being a renewable resource (plant).

Typically for the kind of galactic civilization you are talking about energy becomes a limiting factor, but your premise says that that is cheap and abundant so its out. Instead lets look at another part of your premise, and use that to drum up some resource demand.

Go-Fast Juice

You cannot have a multi-galaxy spanning empire without the ability to get places quickly. Thus, your hyperdrives provide the perfect excuse for us to have a constant, high demand for a resource. All you have to do is say that that hyperdrives require massive amounts of energy(cheap) as well as some other physical material to work. This other material has to be something that cannot be artificially created with any technique that we know about now, but that is a much smaller handwave than the fact that it lets you travel across the galaxy in a week.

Any ship that uses a hyperdrive will need some amount of this resource, and if you make it so that it wears out over time then there will be a constant need for it. Without hyperdrives the whole galactic civilization grinds to a halt, so this resource will be the one of the most important things to have a supply of. Added together and you have the perfect recipe for an expensive material.

Handwavium Examples

Here are just a few quick ideas for what this material actually is, and why it is rare/expensive.

Crystal Handwavium: The crystal that lets your hyperdrive work can only be found in specific areas/conditions. It forms naturally over time, meaning the supply is also limited. The crystal structure is too complicated to easily replicate. (A breakthrough in artificial crystal making is a good way to shake up the setting later on)

Refined Handwavium: The engine of your hyperdrive needs to be made from a very specific alloy which requires lots of processing and refining of different base resources. While none of this is expensive by itself, in aggregate the amount of time and resources that go into each gram of alloy is enough to justify the high cost of the final result.

Rechargeable Handwavium: Your hyperdrive requires a massive influx of energy which would be unsafe to generate using antimatter alone. Instead, energy is stored in large, rechargeable batteries that can then be dumped all at once to provide the necessary spike. The battery itself is not that expensive, and neither is the energy that goes into it. In this case the cost is simply due to the time it takes to charge each battery. Each hyperdrive trip can use up weeks, months, or even years worth of energy to perform. There is a maximum charge rate that effectively makes it so that users are better off just selling empty batteries to a charging facility and buying a new battery for the next trip.

For each of the examples above, all you have to do is tweak whatever increases demand or decreases supply and you can change how expensive it is to whatever suits your needs. Make the crystal only come from a single planet, or each battery require year's worth of energy for even short trips, or the refined alloy decay at a rapid rate and in constant need of replacement. You are already giving this setting a cool hyperdrive system, you might as well make them pay for the privilege.

Handmade items made by a famous artisan from a famous planet.

These items could be decorative (paintings, vases), utilitarian (glassware, pottery, handbag, shoes), or functional (instrument, book (signed and numbered).

Basically anything that takes personal effort to make will cost more simply because someone will have to put forth the effort rather than just ordering something up from the fabricator.

A similar issue would be a subscription to WoW or other MMO game from a famous company or personality.

Using a Nuclear Reactor or Particle Accelerator, it should be possible to synthetically create any element.

Given that any element can then be made, the most expensive material will be determined by only a few factors.

• Cost of production. Think electricity, staffing costs, and so on.




• Transportation costs. If you need an unstable isotope that can't be kept in storage, you will need the material to be made very close to you. This is because it will not survive a long journey. Having to set up a lab nearby to create the material, or waiting for someone with the ability to drop by, may elevate the cost of said material.




• Supply and demand. In the end, as Joe Bloggs pointed out in his answer, this is the biggest determinant of cost. If you have a rare material that no one wants, it may be cheap. But a plentiful material that is always in short supply may be the most expensive. The most expensive resource will be what your civilization needs the most.

Some ideas for expensive materials:

• Heavy metals with short half lives.




• Carbon, Oxygen, Silicon, anything that might be needed constantly.

Some great reading on the synthesis of elements, cited by Agrajag in the comments:

All in all it depends on what materials they use the most of. If the galactic strip miners use more carbon than they produce then you can bet it will become expensive in very short order! However, if they’re pulling apart the galaxy as fast as I think they are: there’s one thing that I think would be of interest:

Lithium

Lithium is incredibly useful. It is also (cosmologically speaking) quite rare. Stars that are good at making it are also good at destroying it within themselves (the so called lithium discrepancy), so the amount that’s hanging around in the universe today isn’t actually that dissimilar from the amount that was made in the first place (in the Big Bang).

Given it’s myriad uses and the fact that you can only expect it to get more scarce as time goes on and your race uses it up the smart super-corporation would buy all the Lithium possible (raising the price), then sell it back later as galactic stocks run low.

Invest in Lithicorp today!

Astatine

From Wikipedia:

Astatine is a radioactive chemical element with symbol At and atomic number 85. It is the rarest naturally occurring element in the Earth's crust, occurring only as the decay product of various heavier elements. All of astatine's isotopes are short-lived; the most stable is astatine-210, with a half-life of 8.1 hours. A sample of the pure element has never been assembled, because any macroscopic specimen would be immediately vaporized by the heat of its own radioactivity.

[...] Astatine is likely to have a dark or lustrous appearance and may be a semiconductor or possibly a metal; [...]

[...] A visible piece of astatine would immediately vaporize itself because of the heat generated by its intense radioactivity. It remains to be seen if, with sufficient cooling, a macroscopic quantity of astatine could be deposited as a thin film. [...]

[...] Astatine is sometimes described as probably being a black solid [...], or as having a metallic appearance (if it is a metalloid or a metal). The melting and boiling points [...] are estimated to be 575 and 610 K (302 and 337 °C; 575 and 638 °F), respectively. [...] Astatine sublimes less readily than does iodine, having a lower vapor pressure. Even so, half of a given quantity of astatine will vaporize in approximately an hour if put on a clean glass surface at room temperature.

^{Emphasis mine}

I think it could be a very valuable resource for rich people and entertainment for parties.

The richest people in the galaxy could use this resource as a new show to entertain guests during special events and parties. Imagine placing an astatine statue (or in another shape) in the middle of a room and show how this fine piece of art melt and evaporate over the time of an hour. Maybe, the party could finish when the last piece of the statue evaporate.

This resource would be extremely expensive since its extremely difficult to gather, and you must:

1. Find the resource or "manufacture" it though polonium-2010 decay or bombard of Or ²⁰⁹₈₃Bi with ⁴₂He.


2. Harvest it.


3. Refine it.


4. Shape it to the piece of art or furniture you want.


5. Deliver it with FTL ships.

Everything in less than 8.1 hours (astatine-210 half-life)!

I can imagine that the richest persons will have to make queue and reserve in anticipation for a find of this precious and exotic resource. Even more, during meetings, the waiting of these pieces of astatine to arrive at the party could be an event itself!

Parties will have to be made with a lot of planning near zones where miners think there could be astatine. Or, scientific should make an extremely fast FTL drive, able to supply the required celerity needed in this product.

Even more, there could be several types of pieces. Some astatine isotopes have half-lives of an hour, other minutes and others even seconds. Each one for a different kind of event!

Nanomaterials

According to your stated technologies, I will guess that Nanomaterials will be the most expensive materials in this society that you describe, depending on the technology that exists in your hypothetical fabricator technology. The structures can be complex, and thus will take complex and powerful fabricators, and will have a production bottleneck. These types of structures are desired for their fantastic properties.

As of 2017, endohedral fullerenes, a nanostructure formed by trapping a larger atom within a carbon fullerene shell, were the second most expensive material on our planet.

Nanomaterials can have many fantastic physical properties.
Carbon nanotubes or sheets are expected to have tremendous physical strength. We have recently developed transparent ceramics using nanoscale techniques. Among these, Aluminum Oxynitride, can be used to make bulletproof windows.

If you don't like that, moving down the list of our current most expensive materials, we find Californium. I wouldn't use that, but would move on down the periodic table, to something within the (hypothetical) island of stability and create a new element. This new element would be incredibly dense, and likely very hard, and would replace things for which we currently use depleted uranium: Armor, Ammunition, and as a tamper in Nuclear Weapons

Since other questions have mostly not explored the meaning of "material", I proceed as follows:

Material

_{Entry 2 of 2 definition 3b}

noun: a performer's repertoire

I suggest that the most valuable material in the whole of the galaxy would be comedy.

Not only extremley popular (being among the most popular forms of entertainment today in America, The UK and much of the rest of the world), but it can be simultaneously entertaining and instructive, it lends itself to cultural critique, and as an audience participates in the mystique of the experience, they can experience a sense of community that the non-religious may not find elsewhere.

As a vehicle for social change:

Stand- up comedy is made unique by the fact that it encourages such a
space, in which people can think critically about the outside space
from within the safety of the liminal or littoral space.

The relationship between the audience and the comedian:

There is the element of conversation in a performance, in which the
comedian delivers her material and the audience reacts, each
adjusting and responding to the other accordingly. Then there is the
balance of the aggression and awe which the audience feels towards
the comedian, who confronts fearful things onstage, but in the process
of doing so breaks social norms. This can be characterized as a
meeting of the sacred and the blasphemous: the audience condones the
stand-up stage as a sacred space, thus allowing the comic to behave
blasphemously there. The audience members feel, to varying degrees, a
combination of awe at watching the comedian take on frightening
taboos, but also feel aggression as a result of the fact that it
offends their social sensibilities.

But how does it relate to social change then?

there is a pedagogy which takes place in stand-up comedy. The act of
experiencing a well-crafted joke couples entertainment with
instruction in a unique way. Because it is not classified as
educational or instructional, stand-up comedy has the potential to
convey 57 messages that often cannot be voiced in other ways. And in
being genuinely entertaining, it has the ability to reach those who
are often not reached by other forms of instruction. The dialectical
nature of stand-up comedy ensures that the best performances are those
in which the humor is based in truth, making it an ideal tool for effectively critiquing unjust or oppressive situations

(The "57 messages" is, I believe a figurative expression of the bullshit that people put up with. The men in white coats are knocking at your skull.)

While referring to comedy, this may be the least funny answer. Irony.

For other refs please see BBC Comic Relief.

Stealing a page from https://www.schlockmercenary.com/ -- post trans uranic elements.

Nuclear synthesis of elements is well within modern nuclear chemistry abilities; given a T3 civilization, mass synthesis of most convential elements should be possible for anything so rare that disassembling entire planets and stars doesn't generate enough of it.

But at the edge of current nuclear chemistry is the synthesis of post trans uranic elements in a hypothesized "island of stability".

An "island of stability" is a region where the proton/neutron shells in a nucleus are more stable than naive models would predict. If we posit that there is an island of extreme stability (not just fractions of a second, but fractions of an eon) in the post-trans-uranic elements, reaching them could be extremely expensive yet feasibly in a T3 civilization.

Nucleosynthesis in stars depends mostly on single-step absorbtion of He4 or other "light" nuclei followed by decay, then more He4 absorbtion. A fast decaying nucleus has little time to "double-absorb" a light nucleaus, so some regions of stable proton/neutron balance are mostly "out of reach".

Some elements are even generated through neutron-star neutron-star merger, like gold.

A hypothetical civilization capable of doing nucleosynthesis on scales more impressive than ramming two neutron stars together might be able to construct elements with useful properties that are otherwise not found in nature.

Of the civilizations technologies, only Warp Drive is beyond our current ken; everything else is just simple extrapolation of our civilization to more fine-grained control and higher energy budgets.

Warp Drive warps space itself. Manipulating near-singularity level gravity gradients to tear apart and merge nuclei could be an example of nucleosynthesis that nature could not duplicate; when it happens (in black hole mergers) it would end too quickly for stable nuclei to form, and any byproducts would cross the event horizon and be trapped inside the black hole.

In aforsaid https://www.schlockmercenary.com/ fiction, PTUs in any quantity are both made by industrial use of gravity generators and key in making gravity generators. To generate a significant amount of PTUs, you first need a significant amount of PTUs. With a slow enough exponential curve (imagine if 1 thousand units of PTUs allows you to generate 1 unit if PTUs per year) and the PTUs themselves being highly useful (if you want warp travel to be cheap without it, imagine if PTUs allow engines that enable warp travel that is 1000x faster), you'd have an extremely, extremely expensive material (suppose the energy output of an entire galaxy is sufficient to generate 1 unit of PTU per year, and 1 unit of PTU is enough to generate a fast-warp drive for a 100 tonne ship).

Suppose there is a factory galaxies. After a thousand years you have 1000 units of PTU, which if gathered up can produce a second unit of PTU per year (ie, every 1000 years, the amount of PTU doubles).

After 10,000 years you'll have 1000 units of PTU produced per year.

After 100,000 years you'll have 1 million units of PTU produced per year.

After 1 million years, you'll have 1 billion units of PTU produced per year.

At any point in this curve, PTUs can easily be the most valuable thing the civilization has; even with an output of 1 billion units/year, a multi-galaxy civilization will find that equipping every one with an unlimited number of starships with "fast" warp wouldn't be trivial. And that could easily be a good chunk of a million years in the future.

If fabrication of anything is cheap, but transmutation of elements is not allowed, then the logical answer is "Material containing chemical elements with the highest demand/supply ratio".

It is not necessarily the rarest of the elements - if this intergalactic society has no use for it, it would be relatively cheap.

But I think it is very unlikely that a Kardashev III civilization would have no ability to transmute elements on large scale. Transmuted elements may have a higher cost (like desalinated water is more expensive than natural freshwater), but they should be widely available.

Anti-matter. It's really difficult to create and to store, but it has the highest energy density of any material, so it'd be very useful as any sort of fuel.

The same things that are expensive now, because they are done the old fashioned way

Organic Food - People today are quite willing to pay a premium for organic food despite questionable evidence that it's much better for you.

Mined Diamonds - People pay a HUGE premium for diamonds mined from the ground, even though even the experts say they'll never be able to see the difference.

I am not certain about how you define materials, but one answer is information. A detailed map of this inter galactic civilization would be invaluable. Also, communications technology from the hardware to the "phone numbers" of people would also be priceless. In a civilization where everything you need is cheap, trade secrets would be one of the few commodities left. Another answer is copyrighted forms of entertainment. An The intergalactic number one hit is sure to be in high demand.

I am not sure a materiel will have much value at all in this kind of super-affluent society, with the whole universe of stuff available.

I think what will have value is:

• A person's time




• A sense of purpose and worthwhile work.

With all physical and material needs essentially unlimited and virtually free, it means people do not really have to work / sell their time.

This means getting someones time and personal service is a big deal.

On the flip side not much reason / need to do anything, so little purpose in life.
For humans this seems to lead to a lot of problems - we (mostly) want to be part of and contribute to society, to do meaningful and valuable work.

This means that the opportunity of doing something important and worthwhile will become a valuable commodity.

Maybe we can combine them and say that the opportunity of performing an important personal service for someone is a valued commodity?
That the chance to teach young children will be something people value highly?

Unstable Reagents

Manufacture is, perhaps, a possibility; but one thing everyone seems to gloss over is that once manufactured, a material would need to remain stable for its use. I encourage you to consider materials like tungsten trifluoride, which is fully capable of continuing to burn in a vacuum and reacts with almost everything. (It's a better oxidizer than oxygen itself, and basically a real-life alkahest; you get it on your fingers, there's little to be done.) The very factor that makes them valuable would also make them extremely difficult to transport.

Another great example is the artificial noble gas, Oganesson. It's element 118, and the most radioactive material known to man. Whether there's a use to it has yet to be seen, as we've only produced it enough times to count on one hand. While your civilization may be able to produce it, it has a half-life of 0.66 ms (more or less), so transporting it would require a handwavium container that's turned up to 11.