If you get 2 dollars on the first head, 4 dollars on the second, etc., then the probability of receiving each dollar amount multiplied by the dollar amount is 1 dollar.

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.