Claude Osteen and Jim Kaat come to mind, but Kaat actually enjoyed something of a renaissance with the Sox, despite Harry Caray constantly complaining that he needed to be released early in 1974. The presence of the DH in the lineup tends to be a magnet to attract players past their prime. The White Sox Bo Jackson was a shadow of the Bo Jackson that came up with the Royals. And Jose Canseco finished his career keeping the DH spot warm for Frank Thomas.
The relievers like Clay Carroll and Mike Jackson who just keep bouncing around at the end seem abundant. Then, of course, you have the grinders slowed by over the years by a few injuries, such as Darin Erstad and Mark Kotsay, whose White Sox careers might have made you forget how good they once were at baseball. You might even put Tony Phillips into that category.
It isn't just the White Sox, of course. Jason Giambi was still in the league the last I looked, and Kotsay hit a pinch-hit home run the other night for the Padres.