The entire play of The Taming of the Shrew—the story about a man who marries a shrewish woman and tames her—is actually a "play within a play". The play opens with a framing scene in which actors are summoned to put on a play. (Once the inner play starts, the framing device is never referred to again.)
The play (and later movie) Deathtrap centers around a fictional play of the same name.
In Pedro Muñoz Seca's theater play Don Mendo's Revenge, exiled Don Mendo reappears in disguise with a troupe and manages a play-in-play reflecting his own, wrongful punishment; some of the people who had wronged him recognize themselves, Hamlet-style.
La Princesse Zenobia and Slaughter on Tenth Avenue in On Your Toes.
The Small House of Uncle Thomas in The King And I.
The Taming of the Shrew is the show-within-a-show for Kiss Me Kate.
Adelaide works at the Hotbox in Guys and Dolls; "A Bushel and a Peck" and "Take Back Your Mink" are actually her performances there.
Cabaret is set almost completely at the Kit Kat Club, a seedy cabaret in Berlin: in the film, all the songs except "Tomorrow Belongs To Me" are performances at the Kit Kat Club, in the original play, "Wilkommen" "Don't Tell Mama", "Two Ladies", "Sitting Pretty" (or in later productions, "Money Money"), "If You Could See her", and "Cabaret" are, while the rest of the songs are not.
Noises Off is a 3-act play about a traveling production of a British comedy called Nothing On. Each act is the same show at different points in the show's run. This type of show is also known as a backstage comedy.
Opera within an opera: Richard Strauss's Ariadne auf Naxos, in its first act, has "the richest man in Vienna" commissioning two after-dinner entertainments: a serious, dramatic opera, and a light musical comedy. Shortly before the first is about to begin, the majordomo arrives and informs the two companies that their sponsor has changed his mind: the two are to be performed simultaneously. The second act is the resulting production.
Louis Nowra's Cosi is a play (and subsequent film) involving the members of a mental asylum rehearsing and (in the final act) performing Mozart's Cosi Fan Tutte.
Anthony Shaffer's play Whodunnit opens it as a bad/obviously satirical parody of an Agatha Christie type mystery (it seems to draw from Christie's Cards on the Table), before revealing that it's actually a play being put on by the characters, who are often very different from their "stage roles".
Marat/Sade is about a group of inmates doing a play.
Man of La Mancha is roughly about a bunch of prisoners performing a play for their own amusement. The production focuses on two storylines: Don Quixote's misadventures, and the "reality" that Miguel Cervantes goes through in prison.
Konstanin's play in Checkov's, "The Seagull"
Vanya's play in Christopher Durang's "Vanya and Sonia and Masha and Spike" is a homage to The Seagull. Its' a hysterically bad story of The End of the World as We Know It from the point of view of a molecule who's also a TV weather person. It also has some parallels with the play itself. Sonia and Vanya discuss what they miss from their old life as themselves (Sonia misses, "her self pity. It was fun.") and it mentions motifs of the blue heron and the wild turkey.
The Crowning Moment of Funny comes when he puts on the record for Act II, and it turns out to be the record for an entirely different show.
Examples of type 3 (SWAS is plot point)
Stage Blood by Charles Ludlam is a little bit extreme in this. The whole plot is an adaptation of Hamlet that takes place during a production of Hamlet, while another show (an experimental play written by one of the characters) is performed in place of The Mousetrap Scene.
Cyrano de Bergerac — "The Clorise", the play presented at the Hotel de Bourgougne in the first act, is a real play (that gets interrumpted at the third line by the protagonist!). The first act is titled "A Representation at the Hotel de Bourgogne", where actors are playing The Clorise. The Hotel de Bourgogne was the oldest and most famous stage in Paris during that time and The Clorise was a real play written by Balthazar Baro, a real author of the time Cyrano de Bergerac is settled on. Balthazar Baro's theatrical output includes the innovative play Célinde (1628), which makes imaginative use of the play-within-the-play, as Edmund Rostand, a french theatre scholar himself, was aware. So the play within a play is a real play that was popular in his time, but now is obscure at the very least. Is even lampshaded (and mocked) by this dialogue:
ACT I, SCENE I: The young man:(to his father) What piece do they give us? The Burgher:The Clorise. The young man: Who may the author be? The Burgher: Master Balthazar Baro. It is a play and a half!...
Nixon In China, where in act II Nixon, his wife, and Henry Kissinger attend a ballet and sing the opinions of the ballet dancers in a manner highly suggestive of a strawman of their own views. Based on an actual ballet staged by Chairman Mao's wife, who yells at the dancers en scene when they mess up.
The Drowsy Chaperone, again. Man in Chair mentions that his mother gave him the record before his father left him. "He didn't leave because of the record, although I'm sure it didn't help."
The Real Inspector Hound is a play about two theatre critics watching a play. The play enters serious Mind Screw territory when the lines between the 'inner' play and the 'outer' play are blurred, though. Moon and Birdboot inadvertently become part of the play in the third act, whilst the characters they replace then take their places in the audience and critique them.
The Murder of Gonzago in Hamlet is performed because it mirrors the suspected crime of Claudius. Hamlet organizes the performance to see how he reacts. The quality of the writing (and, in most competent performances of Hamlet, the acting) in The Murder is much less naturalistic than that of the "outer play", to the point of seeming truly stilted to modern audiences. But there's a genius even in the wince-worthy lines: it's a remarkably spot-on imitation of the writing style of the pre-Renaissance morality plays put on by that type of strolling players.
Taken to extremes in Rosencrantz and Guildenstern Are Dead, which makes Gonzago even closer to the plot of Hamlet, including the deaths of two old friends of the young prince, which in the outer play hasn't happened yet. Ros and Guil recognise there's something familiar about the situation, but don't make the connection. In Rosencrantz and Guildenstern, Gonzago even has a play within a play!
Canio, protagonist in Leoncavallo's opera Pagliacci and a commedia dell'arte actor, resolves not to submit to the same fate as his show-within-show character Pagliacci: Canio will not allow himself to be cuckolded and humiliated, and the end, he brings on stage his revenge against his wife and her lover.