This seems like such an obvious result, I imagine that there's extensive discussion of it within the game theory literature somewhere. If anyone has a good paper that would be appreciated

This appears to be strongly related to the St. Petersburg Paradox - except that the prize is in utility instead of cash, and the player gets to control the coin (this second point significantly changes the situation).

To summarise the paradox - imagine a pot containing $2 and a perfectly fair coin. The coin is tossed repeatedly. Every time it lands tails, the pot is doubled; when it eventually lands heads, the player wins the entire pot. (With a fair coin, this leads to an infinite expected payoff - of course, giving the player control of the coin invalidates the expected-value calculation).

Pre-existing extensive discussion probably references (or even talks about) the St. Petersburg Paradox - that might be a good starting point to find it.