In rock and metal, usually the best guitar solos were improvised on the spot, and it's practically impossible to recapture that moment the same way again.
There's also a lot of great improvised moments in movies, and it's always fun to learn what wasn't in the script someone else brought to the film.
The Daily Beast just compiled a list of "Great Improvised Movie Scenes," many of which I knew were improv'd, and a few I didn't realize were made up on the spot.
One of the most famous ones as most people know is Robert DeNiro's "You talkin' to me" routine in Taxi Driver. Director Martin Scorsese always allows a lot of improv in his movies, like his hero John Cassavettes, who improvd most of his films.
The scene in GoodFellas where Joe Pesci's mother makes dinner for everyone in the middle of the night while the guy's dying in the trunk was almost completely improvised, and they also listed the "What do you mean I'm funny" scene from Fellas as well. You can see the real confusion on Ray Liotta's face because it wasn't in the script, and I've heard two different versions of the story where Joe Pesci got the idea from, where he'd either saw a tough guy do that to somebody, or that a tough guy actually did that to him once.
Another famous improv was the "You're gonna need a bigger boat" line in Jaws, and the sound had to be redone after the first preview, because the audience was still going crazy from the shark popping up and couldn't hear it.
Other improvs that went down in history listed by the Beast include Dustin Hoffman screaming "I'm walkin' here!" in Midnight Cowboy, the hilarious line up scene in The Usual Suspects ("Hand me the keys you…"), "You know how I know you're gay" from the 40 Year Old Virgin, Hannibal's Chianti and fava beans line in Silence of the Lambs, John Belushi in the cafeteria in Animal House, "I'll have what she's having" from When Harry Met Sally, and another horror favorite, "Here's Johnny!," from The Shining. And let's not forget the improv every self respecting geek knows, Indiana Jones shooting the show off swordsman in Raiders of the Lost Ark.
Several improvs I didn't know where improvs include Rutger Hauer's "Like tears in the rain" line from Blade Runner, Malcolm McDowell signing "Singin' in the Rain" in A Clockwork Orange (Before he was signing it, Kubrick thought the rape scene was "too stiff."), and one of the most memorable lines in cinema history, "Here's lookin' at you kid." (Bogart would say often this to Bergman when he was teaching her how to play poker.) And a fun little bit of trivia I never knew, apparently DeNiro got "You talkin' to me" from Bruce Springsteen, who used to say that you his audiences when they'd chant his name.