You're immortal. Tell Omega any real number r > 0, and he'll give you 1-r utility. On top of that, he will give you any utility you may have lost in the decision process (such as the time wasted choosing and specifying your number). Then he departs. What number will you choose?

This is rather tangential to the point, but I think that by refunding utility you are pretty close to smuggling in unbounded utility. I think it is better to assume away the cost.