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That looks like the St. Petersburg Paradox. Much ink has been spilled over it.
The expected payout is infinite. At any point, the "rational" (profit-maximizing) decision is to keep flipping, since you wager a finite sum of money to win an infinite sum. It's very counter-intuitive, hence called a paradox.
In reality, a casino has finite money. You can work out how many coin flips it takes to bankrupt it. So you can work out how likely it is to reach that point with a given, finite sum of money. Martingale strategies have already been mentioned.
Not quite the same, since in my scenario the player loses everything after a loss while in the St. Petersburg Paradox it seems they keep their winnings. But it does seem relevant in explaining that expected value isn't everything.