You are in hell and facing an eternity of torment, but the devil offers you a way out, which you can take once and only once at any time from now on. Today, if you ask him to, the devil will toss a fair coin once and if it comes up heads you are free (but if tails then you face eternal torment with no possibility of reprieve). You don’t have to play today, though, because tomorrow the devil will make the deal slightly more favourable to you (and you know this): he’ll toss the coin twice but just one head will free you. The day after, the offer will improve further: 3 tosses with just one head needed. And so on (4 tosses, 5 tosses, ….1000 tosses …) for the rest of time if needed. So, given that the devil will give you better odds on every day after this one, but that you want to escape from hell some time, when should accept his offer?
I haven’t worked through this one formally, but I have the sinking feeling that the correct answer is to choose an awfully long (infinite?) period of torment. Think about it. Waiting another day adds only slightly to suffering, viewed as part of a potentially very large total. But you improve your odds of escape by a considerable amount. Your best chance of getting out of the paradox is to have a very high discount rate and a very low level of risk aversion, noting of course that under some utility functions this combination of features cannot be made to fit together.
By the way, if you are into this kind of thing, Will Baude has an excellent post explaining the St. Petersburg Paradox.