Avenue Q features the same actress playing Kate Monster and Lucy the Slut, the same actor as Princeton and Rod, and the same actor as Nicky and Trekkie Monster. The conversations with themselves are particularly impressive because all the puppeteers are onstage and usually one of the puppets is being controlled by a completely different puppeteer who had to match the mouth movements to the other actor's words. Also, they sing too. Possibly lampshaded when you notice that the voice of Nicky/Trekkie Monster has the only puppets that require two people to work (can be used by one, but not as effective).
Charles Ludlam's The Mystery of Irma Vep heavily invokes this trope (and occasionally lampshades it in the script). It's a show with four male characters and four female characters, with two male actors playing all of the roles. In particular, the actor playing Nicodemus and Enid not only holds conversations with himself during costume changes, but also plays out a scene involving playing the monster that's mauling Nicodemus off-stage.
This was commonly done in-story in Shakespeare's plays by his "clown" characters (played by comic actors who were presumably able to do multiple voices):
Feste in Twelfth Night stands outside the dark cell in which Malvolio is imprisoned and enacts a conversation between himself (Feste) and "Sir Topas", the minister supposedly sent to "cure" Malvolio of his insanity. Malvolio falls for it, of course.
In The Wizard of OzOn Ice, Bobby McFerrin voiced all of the characters except for Dorothy. (Yep, even Toto.) In the TV special, he also voices Dorothy.
In 1600 Pennsylvania Avenue (and the concert redux, A White House Cantata), one actor plays every President and one actress plays every First Lady. This makes "Duet for One" interesting, where the actress has to switch back and forth between Julia Grant and Lucy Hayes.