Poll on risk aversion: a continuation of the Bankroll on AA thread

[Derek in NYC]

I started a different thread, but Sfer had a great idea, which I decided to make into a poll. Assume the following:

Your entire bankroll must be at stake. If you lose it, you will be unable to continue to play poker at the level you are accustomed to, and must rebuild.

Bill Gates offers you a proposition bet. You think of a whole number between 1 and X, inclusive. Gates guesses. If you win, you double your bankroll. If you lose, you wait for July's rakeback and start over. How big does X have to be? (The question essentially asks you to define the odds you require to risk your bankroll.