So I don't really know how utilons work, but here is an example of a utility function which is doubling box-proof. It is bounded; furthermore, it discounts the future by changing the bound for things that only affect the future. So you can get up to 1000 utilons from something that happens today, up to 500 utilons from something that happens tomorrow, up to 250 utilons from something that happens two days from now, and so on.

You are right that such a utility function cannot be supplied with a (utility) doubling box problem---and for much the same reason that most utility functions that approximate human preferences could not be exposed to a doubling box. Nevertheless this amounts to refusing to engage with the game theory example rather than responding to it.