This seems like a helpful simplification of the problem. Note that it also works if you receive 1-1/N utilons, so as with the original post this isn't an unbounded utility issue as such.

Just one point though - in the original problem specification it's obvious what "choose an integer N" means: opening a physical box on day n corresponds to choosing 2^n. But how does your problem get embedded in reality? Do you need to write your chosen number down? Assuming there's no time limit to writing it down then this becomes very similar to the original problem except you're multiplying by 10 instead of 2 and the time interval is the time taken to write an extra digit instead of a day.