Alice and Bob play a game in which each flips a fair coin repeatedly until it turns up tails, earning a score equal to the number of times it turns up heads. (Thus if Alice flips $HHHT$, her score is $3$.) The high scorer wins, and collects a prize of $4^n$ from the loser, where $n$ is the loser's score.
Let $P(a,b)$ be Alice's payoff when she and Bob earn scores of $a$ and $b$. Then Alice's expected payoff is $$\sum_{b=0}^\infty{1\over 2^{b+1}}\sum_{a=0}^\infty {1\over 2^{a+1}}P(a,b)=1/2>0$$ so that Alice, if she's an expected-value maximizer, should certainly agree to play this game.
On the other hand, her expected payoff is also $$\sum_{a=0}^\infty{1\over 2^{a+1}}\sum_{b=0}^\infty {1\over 2^{b+1}}P(a,b)=-1/2<0$$ so she should certainly refuse to play.
Some further words on the matter (including a reference to a MathOverflow post!) can be found here.
