This is a staple of Theater of the Absurd , especially Eugene Ionesco.
Aria da Capo, a one-act play by poet Edna St. Vincent Millay is interesting in having only three scenes, the first and last of which are nearly identical. In music "Da Capo (al Fine)" means "return to the start and play again until the "Fine" (an indicator in the music where it should end).
God (A Play) by Woody Allen ends with a closed loop - the dialog is the same as the beginning, it is suggested that the play could go on forever (like The Song that Doesn't End).
Wicked begins and ends with mostly the same scene ("Good news! She's dead!"), but the tone is very different.
Sunday In The Park With George begins with an artist musing about the blank piece of paper on which he is about to start sketching: "White. A blank page, or canvas." A hundred years later, his great-grandson sets out to create a new piece of art, and ends the musical with the exact same words.
The first and last words in Into the Woods are "I wish", sung on the exact same notes. The play opens with the Narrator saying, "Once upon a time, in a far-off land..." And the last words spoken (not sung) is the Baker saying those same words to his own son.
Orff's Carmina Burana begins and ends with "O Fortuna."
In Parade, the finale reprises "The Old Red Hills of Home". In the intro, a young man sings about going off to fight in the American Civil War, and in the finale, a young man sings about going off to fight in the "Great War", showing nothing has changed for the next generation.
"Dites Moi" points out what has changed since it was first sung in South Pacific.
No Strings begins with "The Sweetest Sounds" to show that the Official Couple has not met up yet, and ends with the same song after they've agreed to break up and forget that they ever met.
Rent begins and ends with Mark and Roger in their apartment, the former narrating as the latter tunes his guitar, right before Collins comes home after some time away. The Movie shows this by having him call from a payphone outside asking for the key. Mark even lampshades this by yelling, "Don't get your ass kicked this time!" as he tosses it from the balconey (Collins was mugged in the beginning). An alternate ending included on the movie's DVD shows that it was originally going to have Book Ends: the movie begins with the lead characters singing "Seasons of Love" on a bare stage, and the alternate ending depicts them singing "Finale B" on the same stage. However, even though that opening still appears in the final film, the ending was replaced because the director felt that seeing Angel return would ruin the emotional impact of his death about 15 minutes earlier.
The overture to Mozart's Don Giovanni begins with a short, somber song fragment that is actually rather boring. The end repeats this song, but with the Don, Leoporello, and the Commander all singing, in such a way that sounds much more awesome.
Leos Janacek's opera The Cunning Little Vixen opens with a frog leaping into the laps of a dozing forester, and ends with the forester returning to the same spot, a frog — the grandson of the original frog — leaping into his laps.
The Solid Gold Cadillac begins with a meeting of the board of directors of the General Products Corporation, and ends with another General Products board meeting, except that McKeever and Mrs. Partridge have replaced the Corrupt Corporate Executives. A little old lady tries to ask a question, but Mrs. Partridge says, "Oh, no! That's how I got my start!" and bangs the gavel to conclude the meeting and the play.
Oklahoma! begins and ends with "Oh, what a Beautiful Morning."
The musical revival of Vanities introduced a fourth act, set at least 10 years after the previous (which makes it 20+ years after the first act), with the women reuniting in the town they grew up in. "Looking Good", one of the closing songs, reprises the title of "Hey There Beautiful", the opening number of the Theatre Works version. The characters also remove their makeup at the beginning of the final act/scene, mirroring the opening scene. The off-Broadway version of "Looking Good" also adds a reprise of the scat-singing intro of "Setting Your Sights".
ACT's seasonal play of A Christmas Carol begins and ends with the company singing "God Rest You Merry Gentlemen".
The play "Porches" begins and ends with the song "On My Porch". Set in the working class railroad city of Altoona, Pennsylvania during its heyday, it chronicles the lives of people in four different houses. The happy sounding "On My Porch" is a bit jarring at the end and a bit bittersweet after the death of one main character, a Slap-Slap-Kiss romance forms, and a boy you keep seeing finds his long lost mother. The message changes from "it's a nice day to watch it from my porch" to "our lives may change but we still have our sense of community", even though the lyrics remain exactly the same and the only difference with the song at the end is that the main character who dies is not participating in the song.
13 has Evan start and begin with pretty much the same line, but much more optimistic at the end.
Amaluna begins and ends with a magic ceremony, set to the song "Come Together". The first celebrates Miranda's transition to womanhood, while the last consummates the marriage of Miranda and Romeo.
Grey Gardens: The Musical concludes with the same scene as the In Medias Res prologue, where Big and Little Edie are listening to the record of "The Girl who Has Everything".
The first and last scenes of The Moon Is Blue have Don and Patty meeting on the observation deck of the Empire State Building, introducing themselves in the first scene, proposing marriage in the last.
The Tender Land begins and ends with Beth dancing by herself.
The final scene of If/Then closely mirrors the opening scene, including a reprise of the opening number; it's particularly noticeable when Beth meets Josh for the first time; it's nearly an exact replay of Liz and Josh's first meeting.