TITLE: Most 'unintuitive' application of the Axiom of Choice?
QUESTION [85 upvotes]: It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis.  Even though all these formulations are equivalent, I have heard many people say that they 'believe' the axiom of choice, but they don't 'believe' the well-ordering principle.  
So, my question is what do you consider to be the most unintuitive application of choice?
Here is the sort of answer that I have in mind.  
An infinite number of people are about to play the following game.  In a moment, they will go into a room and each put on a different hat.  On each hat there will be a real number.  Each player will be able to see the real numbers on all the hats (except their own). After all the hats are placed on, the players have to simultaneously shout out what real number they think is on their own hat.  The players win if only a finite number of them guess incorrectly.  Otherwise, they are all executed. They are not allowed to communicate once they enter the room, but beforehand they are allowed to talk and come up with a strategy (with infinite resources).
The very unintuitive fact is that the players have a strategy whereby they can always win.  Indeed, it is hard to come up with a strategy where at least one player is guaranteed to answer correctly, let alone a co-finite set.  Hint: the solution uses the axiom of choice.

REPLY [4 votes]: I know a (perhaps even more counter-intuitive) "game" similar to the one presented in the question. There are 100 people and a room with countable many boxes (numbered by naturals). There is a real number in each box. These 100 people can prepare a strategy and then they are separately going to the room. When one comes to the room he begins to open some boxes. He is allowed to (for example) open infinite number of boxes, then decide which box will be opened next. But on the end there have to remain exactly one closed box. The visitor makes a tip, what number there is in it, and go away. He, of course, can't hint to others. Then all boxes are closed and another visitor comes.
There exists a strategy such that 99 of them gives the right answer.
Hint: The core idea is the same as the one in the problem presented in the question but there is one more step ;-)