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Is there any word for a person who moves frequently between two extremes in everything? For example, sometimes he eats too much and sometimes too little, or sometimes won't stop talking and sometimes won't say anything—like that, moving between two extremes in everything.

A good description is the idiom "She runs hot and cold," (which indicates extremes of mood and behavior.)

all-or-nothing [awl-er-nuhth-ing] adjective -dictionary.com

1. not allowing for qualification or compromise; either fully or not at all operative: an all-or-nothing approach.

Having asked myself this question, I've found that the acronym AON hasn't really caught on yet...

I'm an aon: a real, all-or-nothing kinda guy. I will haz cake and eat it too, or I don't want any at all.

I'd suggest:

"Mercurial", which Merriam-Webster defines as "characterized by rapid and unpredictable changeableness of mood".

"Capricious", which Merriam-Webster defines as "changing often and quickly; especially : often changing suddenly in mood or behavior".

"Volatile", which Merriam-Webster defines as "likely to change in a very sudden or extreme way".

(Edited to add attributed definitions, in accordance with commenter recommendations, and to remove "inconstant", which someone else already mentioned.)

"A person of extremes"

extreme - (noun) "either one of two opposite conditions, feelings, positions, etc., that are thought of as being far from what is normal or reasonable" Merriam-Webster

"Physically he was a person of extremes." Factual Nonsense

"Those who have known me have always chided me for being a person of extremes." Project Bible Truth

"It is said that Gaudi was motivated by architecture...He was a person of extremes." Styles of Organizing

You might also use one of the adjectives: unbalanced, unstable, inconsistent, inconstant, but these do not necessarily mean moving from one extreme to the other.

John Farmer & William Henley, Slang & Its Analogues (1904) offers this saying to describe a person who tends to be very pleasant or very unpleasant:

He is all honey, or all turd.

A much older version of the expression appears in Francis Grose, A Classical Dictionary of the Vulgar Tongue (1788), describing a relationship that alternates between extremes of lovey-dovey bliss and unrepressed rage was

It is all honey or all t——d with them ; said of persons who are either in the extremity of friendship or enmity, either kissing or fighting.

On the single-word front, one option is oscillator, which Merriam-Webster's Eleventh Collegiate Dictionary (2003) conveniently defines as "one that oscillates." The Eleventh Collegiate does better in offering a relevant definition of oscillate:

oscillate vi (1726) ... 2 : to vary between opposing beliefs, feelings, or theories

and it offers these relevant definitions of oscillation:

oscillation n (1658) ... 1 : the action or state of oscillating 2 : VARIATION, FLUCTUATION ... 4 : a single swing (as of an oscillating body) from one extreme limit to the other

I think the act of doing so is to vacillate.

From Merriam-Webster:

vacillate: to repeatedly change your opinions or desires

And so one who does that is vacillating or a vacillator. There isn't a great adjective form though; he is a vacillative guy is probably a stretch.

How about unreliable? This is in the sense of a person who lacks character and their behavior is difficult to predict in all situations?

Sounds like an extremist to me. ;)