The expected value is indeed infinite, but I don't think it follows from that that one should pay an unboundedly high amount to play the game.
Sure, over an unbounded number of games, I can expect to make an unbounded amount of money, however that's only if I'm able to play an unbounded number of games. It doesn't take into account how quickly my cash on hand depletes, preventing me from playing further.
For a fixed amount $n I pay to play, my chance of making a profit in a single game is less than 1/n. (If n is a power of 2, then the chance of profit is 1/(2n), if I n is not a power of 2, then the chance of profit is 1/k where k is the smallest power of 2 larger than n). That's just a profit on the one game though. The chance that I make an overall profit, however, is going to be tricky to analyse I'm going to have to think about that for a bit.
So, I guess it comes down to how many times I am allowed to play. Only once, then $2 (the 50/50 amount, or maybe $4 on a gamble). Multiple times: then it would be an optimisation of how much money I had and how many times I could play vs chance of making an overall profit vs how much I stand to lose.
If I'm allowed unlimited store credit, then maybe I keep playing until I'm on top (or I die, whichever comes first) :) Although, in reality, if somebody was offering unlimited credit on a game like this, I'd expect some sort of scam/trap/trick.