1/2 * 2 + 1/4 * 4 + 1/8 * 8 ... = 1 + 1 + 1 ... = ∞
Of course, the proof only works if dollars continue to have linearly increasing value to you as you keep accruing them. If their value to you logarithmically increases, the expected value is not infinite.
We don't all know that. There's an ever-decreasing but nonzero probability that you will keep getting heads.
But the probability that the next flip is heads is .5 no matter how many of the previous flips were heads :)
But with enough flips, the odds getting heads on all of them comes close enough to zero that they are effectively zero.
But the reward is not limited there. Once you reach each and every "balance point", the next step is 50% chance for doubling the reward. If you currently have X, the value of continuing to play is 1.5X. This is independent of how big X is, or how unlikely you have made it this far (sorta a reverse Gambler's Fallacy). And it's why the expected value is infinite.