>I'm not looking at averages/medians at this point, simply probabilities of winning.

The point is that a small probability of winning can be offset by a large jackpot - as in the lottery (which, of course, is stilled rigged in favour of the organizer). Therefore, I think it makes sense to try to calculate the "value of the game" as sum(probability * amount won), summing all possible combinations. The calculation I presented at first is the simpler approach for this calculation.

You can consider the "infinite amount" as a limit. For instance, assume that the game stops after the coin is thrown the third time, then calculate the average of 8 games - assuming that the first "head" is reached in 4 out of 8 games, the second "head" in 2/8 games, and the third, in 1/8 games.

Now, extend your calculation to 4 throws (assume the game ends here, even if "heads" appears 4 times). Take the average of 2^4 = 16 games.

Now, extend your calculation to 5 throws. Do your calculations for 2^5 games.

In each of these cases, the value of the game is finite, of course. But if you continue...