I'm not sure if this is the most, but Cantor's Diagonal Argument[1] is the first that comes to mind, probably even more as an example of a creative approach than for its implications (although those are pretty huge too).
Charles Petzold's "Annotated Turing" has a great run down of this and a bunch of the other mind-blowing things that came out of math in the late 19th/early 20th centuries.