I guess I didn't formulate the rules clearly enough--if the coin lands on tails, you exit with the stake. For example, if you play and the sequence is HEADS -> HEADS -> TAILS, you exit with $4. The game only ends when tails is flipped.
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.
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