"Non-converging expected values" - you can't conclude that the scenario is contradictory just because your tools don't work.

As already noted, we can consider the problem where you name any number less than 100, but not 100 itself and gain that much utility, which avoids the whole non-convergence problem.

"This would disintegrate the OP entirely" - as already stated in other comments, claims that my situation aren't realistic would be a good criticism if I was claiming that the results could be directly applied to the real universe.