“Dreidel Dreidel Dreidel” is also sung by the cast of “Hairspray” on the Broadways Cares/Carols for a Cure 2004 CD. Harvey Fierstein is great!
The song from Phantom is “Notes.” I love it!
I love stuff like that. You’re talking about canons, fugues, and other very-structured musical forms which formed the backbone of J. S. Bach’s work (as well as the other Baroque composers).
From what I recall, to write songs like that is to perform a very mechanical function. To write songs like that well is to achieve some of the highest pinnacles of Western Art.
Give some Bach a whirl. If you can’t dig that, then you just don’t like counterpoint. No biggie.
And Forbidden Broadway Volume 2 has “Fugue for Scalpers,” a parody of “Fuge for Tinhorns” that is so good.
I got two tickets here
For Phantom in the rear
But you can see real good through the chandelier.
Half-price, half-price, my wife kept these tickets on ice.
This is one of my favorite musical tunes also. A little different than the OP’s point because it doesn’t recombine previously separate parts; separate parts are woven together through the entire song.
However it is called Prima Donna. “Notes” immediately precedes it; they kind of run together.
the technical (musical) term for singing/playing two or more different melodies at the same time is called
(drum roll please)
Quodlibet
Count me among those who like it too. A few other examples:
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Also from the Buffy musical, near the end of ‘walk through the fire’ there are, I think, parts where three different people singing at once. (Buffy, Tara, and Sweet?) Certainly Sweet sings about a verse under the other characters, which did kind of drive me crazy in a good way until I could figure out what words and tune he was singing.
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‘I, Don Quixote’ from Man of La Mancha, with Sancho and Quixote singing their themes at once and the two melodies just seem to blend together. I remember my mom and I singing along, each taking a part, while the record was playing.
on preview: thanks for the link anyrose
next time you’re in a big enough group - try these two together
Oh When The Saints
and
Swing Low Sweet Chariot
And change the tempo of one, no? I’ve never done this, but I’m trying to “sing” them both in my head, and it sounds interesting…
Also, I believe the end of Act I of The Producers copies the West Side Story device mentioned above (just as South Park did).
Oh! And there was that Simon and Garfunkel thing where they sang Scarborough Fair along with … another song… don’t remember which one.
Also, listen to “Five for One” at this link: http://www.isound.com/albums/bob_ricci/735169/not_a_christmas_album
There’s also the famous “One Day More” which is quite nicely done and stirring to boot.
There’s a third song called “I’m gonna sing” that melds with those two. (Or at least, I’ve always sung all three together)
Scarborough Fair is traditional, and Paul Simon wrote the counterpoint “Canticle”.
Lyrics here.
And there’s one from Call Me Madam: “I hear singing…” or whatever it’s called.
There are two really, really great double choruses in Gilbert & Sullivan, neither of them Act 1 finales; “When the foeman bares his steel” from The Pirates of Penzance and “Tower warders under orders” from Yeomen Of the Guard. 
Another great example is “Quartet” from Chess. Though as someone mentioned my very favorite earlier (“Fuge for Tinhorns”).
This is a great musical and dramatic device – whether individuals like the results or not – it works very well in the structures I’ve seen it used.
Great minds… 
I was beginning to think I was the only one who liked that song. The person who introduced me to phantom hated it.
I meant to mention “The Book Report” from You’re a Good Man Charlie Brown.
Also, just for fun, listen to PDQ Bach’s “The Art Of The Ground Round” to truly appreciate this kind of writing.
My favorite example of the is the “Lover’s Quartet” from Co-ed Prison Sluts. Although it is heightened by Fluffy tearing the luncheon meat intestines out of the “hamster” while Hamster Man sings “The dog has wasted my hamster now, the agony’s wasted my heart.” Sigh. God, I miss that show.
Jack
Peter Gabriel-era Genesis used this structure with Fountain of Salmacis, and I’m thinking that Yes did it as well (Siberian Khatru, perhaps?)
One of my favorite moments from Miss Saigon is when Kim and Ellen are both singing over each other about Chris and their relationship with him.