The fastest I've ever seen in person (rear wheel, mini pump) was in the range of 5 or so minutes.

I was at the back end of a group ride during the warm up when this guy's tire blew. So, I figured I'd wait up for him while he fixed it. However, I'm pretty sure his tire originally blew because the tube was improperly installed to begin with, so that made me less impressed with his speed.

Of course after this fix we were way behind the group and this putz tried to make up ground by hammering 25mph down a bike path to a meet up spot. I let him go on his merry way at that point...