If your utility function is bounded and you discount the future, then pick an amount of time after now, epsilon, such that the discounting by then is negligible. Then imagine that the box disappears if you don't open it by then. at t = now + epsilon * 2^-1, the utilons double. At 2^-2, they double again. etc.

Perhaps I am misinterpreting you, but I don't see how this scheme is compatible with a bounded utility function. For any bound n, there will be a time prior to epsilon where the utilons in the box will be greater than n.

When you say "At 2^-2...", I read that as "At now + epsilon * 2^-1 + epsilon * 2^-2...". Is that what you meant?