First, one that would never have happened because two of the participants by all accounts despised each other. In 1974 Frank Zappa, Lou Reed, and Dion collaborated on a doo wop album. Musicians included Zappa’s “Roxy and Elsewhere” band featuring George Duke, Ruth Underwood, Napoleon Murphy Brock, and Chester Thompson.
Well, if you’re allowing for the possibility that it could still happen, I think it would be cool to have a supergroup with Adrian Belew, John Zorn, Les Claypool, and George Hurley. I don’t know what it would sound like except that I’d expect it to be weird and noisy.
Jack Black and Jack White need to put out a Jack Gray album.
If we’re going with the impossible, how about Mozart and Tool?
I’ll say that it should have been possible at some point; participants should have been active at the same time.
Noel McCalla was a backing vocalist for the band Sniff ‘n’ the Tears and later joined Manfred Mann’s Earth Band. He did all the vocals on Mike Rutherford’s ( of Genesis and Mike + The Mechanics) solo album “Smallcreep’s Day”. Mike knew him from when they were auditioning singers for Genesis after Peter Gabriel left.
When I listen to “Smallcreep’s Day” I often think about what Genesis would have been like if they’d hired McCalla and Phil Collins had stayed behind the drums. I also sometimes wonder what Genesis would have sounded like if they’d hired Roger Taylor instead of Collins. (Taylor wound up getting a pretty good gig later with Queen…)
After Jim Morrison died, the remaining members of The Doors briefly considered asking Iggy Pop to join the band, before deciding to just try moving on as a trio. That would have been an odd combo, and one that I’m not at all sure would have worked, but it would have been interesting to see what came out of that collaboration.
ETA: heh, I just bothered to read all of my own article link and it says Ray Manzarek said in an interview that they considered several lead singer replacements, including Paul McCartney (“hey, we’d have a singer and a bass player”). Talk about what an oddball pairing that would’ve been.
In 1981, on the heels of the breakups of Yes and Led Zeppelin (the latter after the death of John Bonham), Jimmy Page (Led Zeppelin guitarist) and Chris Squire (Yes bassist) began working on an idea for a new band, which they titled “XYZ” (for “eX-Yes-Zeppelin”).
They were joined by Yes drummer Alan White, and a keyboardist named Dave Lawson. They attempted to recruit Robert Plant, but Plant was still in mourning for Bonham, and didn’t care for the complexity of the music which Squire was writing.
Due to not having a lead singer, questions about management of the group, and Squire and White forming a new band, Cinema (which eventually became a reformed Yes), nothing ever came of XYZ, though some of what Squire had written for XYZ made it into Yes’s 90125 album. But, it’s interesting to consider what sort of sound Page, Squire, and White would have created.
In 1970 there was a rumor about Jimi Hendrix getting together with ELP and jamming. This was months before ELP actually formed and released their first album.
Obviously, it never happened. Emerson was quoted somewhere as saying “we can’t play with Hendrix. He’d be louder than I am!”
Sure would have been fun to see, before the inevitable meltdown!
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And, in the “I wish I’d have known about it at the time…” category:
A couple of years ago, I’d noticed that Jeff “Skunk” Baxter had released a new solo album. Baxter was among the first musicians in Steely Dan and the Doobie Brothers back in the '70s. He’s also been a session musician on literally hundreds of other recordings.
As I was listening to his new recordings, I looked him up on Wikipedia and discovered he’d played in a short-lived band in the early 90’s.
They played a short tour of Japan in 1990. Wikipedia shows them as being active from 1990-94. I don’t believe they released any studio recordings.
I give you:
This is the 1st of 9 live concert videos on Youtube. If you don’t want to watch the video, here’s their Wikipedia entry:
Get a load of the players in this group!
Sadly no longer possible-Weird Al and Tom Lehrer.
Throw in Allen Sherman and you got Sherman, Yankovic, and Lehrer. Kinda like Crosby, Stills and Nash with a sense of humor.
I think Sherman is too bland to fit in. As nice as W.A. is in real life, he’s got some (hilariously) dark songs.
Back in the 60s Peter Yarrow (of Peter, Paul and Mary) was putting together an experimental film and asked Tiny Tim to perform a few songs for it. Tiny Tim was backed by a band called the Hawks, who would soon change their name to The Band. Can you imagine Tiny Tim in The Band? 
David Lindley and Bob Brozman.
Stevie Nicks had apparently wanted to join Tom Petty and the Heartbreakers for years, and is said to have made several overtures to Petty, which were rebuffed, apparently because “there are no girls in the Heartbreakers.”
Despite that, Petty and Nicks became good friends, their duet on “Stop Draggin’ My Heart Around” was a big hit (and a great song), and the band eventually considered her “an honorary Heartbreaker.” But, here, too, it would have been fascinating to have heard a more formal, ongoing collaboration.
I hope nobody minds if I give an example of an unlikely musical collaboration that did happen.
In 1971 an English talk-show host, Michael Parkinson, had invited two top violinists onto his show - Yehudi Menuhin (classical) and Stephan Grapelli (jazz.)
As far as I know, they had never played together before - but the result was magical! 
I was going to vote Mozart + Beethoven
How about James Brown with Luciano Pavarotti?
The Masked Marauders, with Bob Dylan, Mick Jagger, George Harrison, John Lennon, Paul McCartney, and others.
It was a hoax written by Greil Marcus for Rolling Stone that should have been obvious to everyone and anyone, considering that he wrote:
“Produced by Al Kooper, the album was recorded with impeccable secrecy in a small town near the site of the original Hudson Bay Colony in Canada.”
“The LP opens with an eighteen-minute version of ‘Season of the Witch’ (lead vocal by Dylan, on which he does a superb imitation of early Donovan). The cut is highlighted by an amazing jam between bass and piano, both played by Paul McCartney.”
“Dylan shines on Side Three, displaying his new deep bass voice, with ‘Duke of Earl’.”
“Paul showcases his favorite song, ‘Mammy’, and while his performance is virtually indistinguishable from Eddie Fisher’s version, it is still very powerful, evocative, and indeed, stunning. And they say a white boy can’t sing the blues!”
“It can truly be said that this album is more than a way of life; it is life.”
But this was 1969 and anything had become possible. And who wouldn’t dream of hearing a supergroup like that? The Traveling Wilburys actually made it happen, sorta.
Leslie West and Pete Townshend sounded great playing off each on the Record Plant version of Baby Don’t You Do It. It would’ve been nice to hear what else they might’ve come up with.