I had what feels like an insight about this type of money pump scenario in general, so I'll run it by everyone.
If X is a painful experience, like a stubbed toe. Y is a second painful experience, like a caffeine headache, and Z is a third painful experience, like a sore muscle, then trading repeatedly in the manner listed above seems entirely correct, since you would essentially be repeatedly trading to eventually almost no pain whatsoever. So the circularity of the trades themselves do not appear to be a problem (as opposed to the circularity of the preferences). As another example of this, if someone is offering 2 bagels for 1 apple, someone else if offering 2 candies for 1 bagel, and someone else is offering 2 apples for 1 candy, and you get utility from apples, bagels, and candies, then you have a great arbitrage situation.
In this case, your preferences are wrong, but they are coherently wrong. If you flip the sign bit, (what you think is good is actually bad and vice versa) for instance, you'll go back to perfectly reasonable, so you can say "If I look three steps ahead, in this scenario I would be better off doing what I would intuitively prefer NOT to do at each step."
This also appears to work whether you flip the sign bit on X's value (Actually you SHOULD discard X because it hurts) or the trade order (Actually, you should make all of those trades in the reverse order then you would think.) but presumably not both, because then you would be voluntarily trading to what you currently think is infinite disutility, which brings up a weird point. Reversing your preferences and reversing how you ACT on those preferences shouldn't do anything to change your behavior, but in this case if you attempt to evaluate the results of doing it in this case, it appears to be infinite disutility. That's probably extremely clear evidence you need to change your value system.
Am I onto something, or am I missing the point?
Edit: Clarified a point about circularity.
Edit: for reasons given in the comments, I don't think the question of what circular preferences actually do is well defined, so this an answer to a wrong question.
If I like Y more than X, at an exchange rate of 0.9Y for 1X, and I like Z more than Y, at an exchange rate of 0.9Z for 1Y, and I like X more than Z, at an exchange rate of 0.9X for 1Z, you might think that given 1X and the ability to trade X for Y at an exchange rate of 0.95Y for 1X, and Y for Z at an exchange rate of 0.95Z for 1Y, and Z for X at an exchange rate of 0.95X for 1Z, I would trade in a circle until I had nothing left.
But actually, if I knew that I had circular preferences, and I knew that if I had 0.95Y I would trade it for (0.95^2)Z, which I would trade for (0.95^3)X, then actually I'd be trading 1X for (0.95^3)X, which I'm obviously not going to do.
Similarly, if the exchange rates are all 1:1, but each trade costs 1 penny, and I care about 1 penny much much less than any of 1X, 1Y, or 1Z, and I trade my X for Y, I know I'm actually going to end up with X - 3 cents, so I won't make the trade.
Unless I can set a Schelling fence, in which case I will end up trading once.
So if instead of being given X, I have a 1/3 chance of each of X, Y, and Z, I would hope I wouldn't set a Schelling fence, because then my 1/3 chance of each thing becomes a 1/3 chance of each thing minus the trading penalty. So maybe I'd want to be bad at precommitments, or would I precommit not to precommit?