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 <a href="http://www.thebigquestions.com/alice.pdf">here</a>.