The classic solution is that $s are not equal to utility. People have some discount rate. But you can trivially rephrase the problem, so the $ increase at the same rate of your discount rate, and you get the same problem.

Only a bounded utility function is a solution - there must be some amount of money where literally even a trillion dollars more doesn't matter.

That seems acceptable, but it still means this game is worth some amount to play, and that amount can still grow very large before reaching your bound. Also there must be some things which we can't bound.

There is a very related problem called Pascal's Mugging: http://wiki.lesswrong.com/wiki/Pascal's_mugging

In Pascal's mugging, a mugger asks you to pay him $5, or he will kill 3↑↑↑3 people (an incomprehensibly huge number, that for all intents in purposes, might as well be infinity.) He says that he is the matrix lord and likes playing games with simulated people.

This is of course, incredibly unlikely. But is the probability he is telling the truth greater than 1/3↑↑↑3? Is $5 worth more than a human life? If so you should pay him.

This is a general problem with expected utility. EU only cares about the average utility. The utility of all the possible outcomes, weighted by their probability. A single outlier can throw the average case off a lot.

EU is forced to trade away utility from the majority of probable outcomes to really weird unlikely outcomes, like the mugger, or winning an infinite series of coin flips. EU is optimal in most everyday problems, but it can fail in extreme cases.