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Mestroyer comments on The Doubling Box - Less Wrong Discussion
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Even better, however, would be to toss two coins every day, and only open the box if both come up heads :)
This suggests a strategy; tile the universe with coins and flip each of them every day. If they all come up heads, open the box (presumably it's full of even more coins).
better yet, every day count one more integer toward the highest number you can think of, when you reach it, flip the coins. If they don't all come up heads, start over again.
That way your expected utility becomes INFINITY TIMES TWO! :)
There are meaningful ways to compare two outcomes which both have infinite expected utility. For example, suppose X is your favorite infinite-expected-utility outcome. Then a 20% chance of X (and 80% chance of nothing) is better than a 10% chance of X. Something similar happens with tossing two coins instead of one, although it's more subtle.
Actually what you get is another divergent infinite series that grows faster. They both grow arbitrarily large, but the one with p=0.25 grows arbitrarily larger than the series with p=0.5, as you compute more terms. So there is no sense in which the second series is twice as big, although there is a sense in which it is infinitely larger. (I know your point is that they're both technically the same size, but I think this is worth noting.)