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Let's say I was at the very center of the enormous Boötes void, way out in deep, deep space. What could I see with the naked eye? I assume I could see no individual stars, but could I resolve any galaxies? If I gazed in the direction of a super-cluster of galaxies would it seem brighter than other directions? How dark would it be compared to, say, the far side of the moon when it is a full moon on earth?

I am told there are, in fact, a few galaxies in the void. So let's say I pick a spot in the void that is as far from any of those galaxies as possible.

The number density of galaxies in a void is typically an order of magnitude lower than the average in the Universe (e.g. Patiri et al. 2006). In this astronomy.SE post, I estimate the number density of galaxies of magnitude $M=-17$ or brighter in the Boötes Void to be $n sim 0.004,mathrm{Mpc}^{-3}$, or $10^{-4},mathrm{Mlyr}^{-3}$ (i.e. "per cubic mega-light-year"). Hence, the typical distance to a galaxy from a random point in the Boötes Void is
$$
d = left( frac{3}{4pi n} right)^{1/3} simeq 13,mathrm{Mlyr}.
$$

That distance corresponds to a distance modulus of $mu simeq 28$, so the apparent magnitude of the galaxy would be
$$
m = M + mu simeq 11.
$$
Typically, humans cannot see objects darker than $m simeq 6.5$ (the magnitude scale is backwards, so darker means "larger values than 6.5"), although some have claimed to be able to see $msimeq8$ — still an order of magnitude brighter than the $m=11$ estimated above. Moreover, this threshold assumes point sources, whereas a galaxy has its brightness smeared out over a quite large area, lowering its surface brightness significantly!$^dagger$. Note also that, as in the rest of the Universe, galaxies in voids are not completely randomly scattered throughout space, but tend to cluster in clumps and filaments, and that the number density is smaller in the center of the void, meaning that here the typical distance to the next galaxy is larger.

Hence, you would be floating in complete darkness.$^ddagger$

$^dagger$_{For instance, the Andromeda galaxy has an apparent magnitude of $m=3.44$ which, if its light were concentrated in a point, would make it easily visibly even under light-polluted conditions.}

$^ddagger$_{Your eye might be able to detect individual photons, at stated in Árpád Szendrei's answer, but that hardly counts as "seeing anything".}

From the Wikipedia page on the Boötes void (the same that you linked to in your question):

According to astronomer Greg Aldering, the scale of the void is such that "If the Milky Way had been in the center of the Boötes void, we wouldn't have known there were other galaxies until the 1960s."

(The original article can be found thanks to the Wayback Machine.)

This means that if we were in the center of the Boötes void, we wouldn't be able to see any galaxies with the naked eye, and not even with some telescopes. Again according to Wikipedia, telescopes were invented at the beginning of the 17th century, radio telescopes in the 1930s and infrared telescopes in the 1960s.

In fact you would see only individual stars. Like on Earth, you can only see individual stars, and no galaxies with the naked eye. Why? Because galaxies are too dim as a whole.

Now, if you would just float in empty space in the void you would be better off then on Earth. It is because here on Earth there is some light pollution.

But in the void, there is basically no light pollution, so you could see even single photons arriving from far away giant stars.

And yes, the human eye is able to detect even single photons.

By the way, all the stars that you can see with the naked eye from Earth are from the Milky Way. But again, that is too because of the light pollution and because our night sky even if there would be no light pollution here on Earth would be filled with brighter start from the Milky way. Our eyes would get used to that, and the start from other galaxies would just appear too dim to see.

Now on the dark side of the moon it would be a little different, but only that there is no light pollution like on Earth. Still, from the dark side of the Moon you could still only see stars from the Milky way.

Now in the void in your case, I think it is different. It is so dark and void, that even a single photon coming from a far away large star would be visible to our eyes that get used to the darkness.

Please see here:

Do all the individual stars that we can see in the night sky belong to Milky Way?