I'm very sure I've seen a couple of fics over the years that do something along this line. Probably Picard/Xavier.
Now I'm thinking I want to glue together a couple of the doppelgangers into a fic in which most of the characters are themselves (not that ST:TOS didn't already play this game with the Mirror 'verse. Hm, and DS9. And to an extent TNG. Ummm. ANYway. But those were same characters, different quantum realities, so not quite the same). Like, hm, Richard Kimble tries to fly the Millennium Falcon. Oh, or president whatsisnamein Air Force One. Okay, now I am very amused by thinking about Han Solo watching him try to fly that plane and being all OH FOR THE LOVE OF GOD. Anyway. But with more than one in the same fic. Ummmm. Picard and Galdalf find themselves in the glass prison cell playing chess when Alex Mead shows up to rescue them. Gilbert Lowell and George Burnett find themselves desperately trying to save a patient in a Chicago emergency room in 1991.
Because I totally need more ideas. I hate you.
Heh. Jaime Escalante shows up to inspire the troops on Battlestar Galactica. Everyone, of course, freaks the fuck out.