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DanielLC comments on Open Thread - Aug 24 - Aug 30 - Less Wrong Discussion
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Also notice that as formulated ("You are given an initial stake of $1") you don't have any of your own money at risk, so... And if the game only ends when TAILS is flipped, there is no way to lose, is there?
If the first $1 comes from you, you are basically asking about the "double till you win" strategy. You might be interested in reading about the St.Petersburg paradox.
The money that's "at stake" is the amount you spend to play the game. Once the game begins, you get 2^(n) dollars, where n is the number of successive heads you flip.