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Let's say you play a game, where you repeatedly flip a coin, and the first time the coin lands tails, you stop flipping. Then you get 2^n coins, where n is the number of heads flips. How much money is the game worth?

The probability of n being 0 is 1/2 (first flip tails).

The probability of n being 1 is 1/2*1/2 (first flip heads, second flip tails).

The probability of n being 2 is 1/2*1/2*1/2 (first and second flip heads, third tails).

And so on.

So, the expected payoff is:

(1/2)^1 * 2^0 + (1/2)^2 * 2^1 + (1/2)^3 * 2^2 + ...

= 1/2 + 1/2 + 1/2 + ...

The expected payoff is infinite, and therefore I should offer any finite amount of money to play this game. Can anyone explain?

Either intuition and a finite number of experiments don't work well with an infinite number of possibilities, or math doesn't.