The probability that player number $i \in \\{1,2,3\\}$ wins is given by $$\sum_{k=0}^\infty \frac{1}{2^{i+3k}}=\frac{2^{3-i}}{7}$$ The reason for the infinite sum (instead of your reasonable attempt) is that there is a probability that the two players other than player $i$ will get only tails and give player $i$ yet another turn, and this could potentially go on forever (although the probability for that scenario goes to zero, which is why the sum converges).
Feel free to ask further questions in the comments below if this argument is not yet clear to you.