Currently, the AL Wild Card standings look like this:
Team W L
Rays 90 61
Red Sox 84 68
White Sox 80 72
At this point, the Rays' magic number to eliminate the White Sox stands at 1: they're guaranteed at least a tie. So if Chicago wins all of their remaining 10 games, Tampa loses all 11 of theirs, and Boston wins 5 or fewer of their final 10, the White Sox and Rays will both finish 90-72 and need a Game 163 to decide the AL Wild Card. Or if Boston wins exactly 6 out of 10, they'll be 90-72 as well, and there'll be a three-way playoff (which would be awesome, not least for the Twins). Of course, the Red Sox can't win more than 6 out of their last 10, because they have four games to play in Chicago--and this whole hypothesis presumes Chicago will win all of their remaining games.
Following the above logic, Baseball Prospectus currently lists the White Sox' chances of getting the Wild Card at 0.00015%, which sounds about right. (ESPN, meanwhile, rounds that off to 0.0%.) I think I'd enjoy betting one cent on Chicago at the Prospectus odds; if I win, I'm up more than $6,500.
Edit added early Friday morning: ...Okay, they're dead. The Rays defeating the Yankees on September 23rd both mathematically eliminated Chicago and ensured Tampa Bay has the tiebreaker (though not, currently, the overall lead) over New York for the AL East title.