My poor feeble meat brain can only represent finitely many numbers. My subjective probability that you'll pay off on the bet rounds to zero as the utilities get big. So the sum converges to less than a dollar, even without hitting an upper bound on utilities.

I think the aforementioned feeble brain would be better off if it only represented utilities up to a certain size too. Imagine you find yourself with a choice between eating an apple and taking an action with a large payoff L and low probability P. If P rounds to zero, then you know that P is between zero and zero + epsilon, so L * P is somewhere you don't know between zero and L * epsilon. If L * epsilon is larger than the expected utility of eating the apple, you won't know which to do, and you don't even know how much utility you might be giving up by taking the wrong choice. In practice you need the maximum possible L to be smaller than the utilities you typically care about divided by epsilon.