A Conundrum

Suppose a generous millionaire offers to play a game with you, at no charge. She will allow you to flip a coin, over and over, until you finally flip heads. (It is a fair coin.)

If you get heads on the first flip of the coin, you will get $1, and the game is over.
If you get heads on the second flip of the coin, you will get $2, and the game is over.
If you get heads on the third flip of the coin, you will get $4, and the game is over.

In general, if it takes n trials to get a head, you will receive 2^n dollars.

Now, suppose that another person taps you on the shoulder. "I would like to play in place of you, and will offer you a sum of money to change places with me."

How much should be offered before you decide to let the other person play in your place?