Infinite bankroll paradox

[Uniqueuponhim]

Alright, I know this problem has absolutely no bearing whatsoever on real life, but I think it's an interesting problem so I'd like to ask what the "solution" is:

Say a man with an infinite bankroll comes up to me and bets me everything I own at 1:1 payout that if he flips two coins they will both turn up heads. Clearly this bet would be +50%EV for me to take. Now let's say that every time I win I have the option of making the bet again, but the man requires that I bet everything including previous winnings each time. Every time I take the bet, my EV is +50%, but if I keep taking the bet over and over again, eventually I'm going to lose.

Looking at it another way, the probability of me losing the bet at least once goes to 1 as the number of bets I make goes to infinity. Therefore I can set my total number of bets to an arbitrarily high number and my overall EV will increase exponentially with the number, but if I take the limit as the number goes to infinity, my EV goes to -100%.

So I have two questions: How do you reconcile this mathematically, and what is the correct (ie highest EV) action with respect to the man's bets? The answer can't be to keep betting infinitely (ie until you lose) because that makes your EV -100%. Yet you also can't set a finite number of bets that would be correct because it's always +EV to make at least one more bet. So what is the solution?