They originally worked as members of the Compass Players, which was the forerunner of the famed Second City. (Nichols also worked as a staff announcer for classical music station WFMT in Chicago, where he created a sort of variety show, "The Midnight Special," which is still on every Saturday night after over 50 years. While at the station, he and May would often record some comedy bits, none of which were ever released on album, but the station periodically plays on the air, from time to time.)
The two did team up again not terribly long ago, when Nichols directed the movie, "The Birdcage," which Elaine May wrote the screenplay for, based on the French film, La Cage aux Folles. The film starred Robin Williams and Nathan Lane.
And here they are teamed up as they were initially, albeit in hiding -- appearing as the Mystery Guests on What's My Line?