An embedded narrative, or a story within a story, is a literary devise in which the character in a story becomes the narrator of another story. Think Canterbury Tales or One Thousand and One Nights. In dramatic literature, the embedded narrative is a play within a play. Shakespeare famously employed the device in Hamlet with the traveling players performing The Murder of Gonzago ("the play's the thing wherein I'll catch the conscience of the king").
Broadway musicals have been known to embed a narrative or two, as well. Sometimes the show within a show is just a short piece, with the characters of the main musical performing only a scene within the larger construct, such as the song "The Grand Canal" in the musical Nine, a scene from Guido's movie musical about Casanova. Other times, like in The Drowsy Chaperone, the framing device is smaller (The Man in the Chair) and the show within the show (The Drowsy Chaperone) is the larger part of what the audience is seeing. Then there are shows like Kiss Me, Kate and Man of La Mancha, which are really blended, giving almost two complete (but not entirely separate) stories.
We've dug through a few librettos and found the fictional titles of a few Broadway musical shows within shows. Some of these might be tricky, but some of them are big production numbers. Click through the gallery below, and see if you can identify the Broadway musical based on the title of the show within it.