Here's an interesting puzzle.

A game is played by repeatedly tossing a coin until it lands heads.  If it 
lands heads on the first try, you win $1 and the game is over.  If it lands 
heads on the 2nd try, you win $2 and the game is over.  On the 3rd try $4, 
and in general if you get the head on the nth try, you win $2^(n-1).

I simulated this game in C++ and after 1e6 goes the average win-per-go 
settles down quite nicely to $8.09 and stays there up to 1e8 goes.  Is that 
correct?  How much should you be willing to pay for each go?  Does it depend 
on how many goes you are going to have?

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