The Match Game Theme Song. Bass, guitar then trumpets 
The Academy Award-winning Theme from “Shaft”!!! (hi-hat, guitar, keyboards, etc.)
My absolute favorite song like this; the crescendo when all the layers have built up to Eddi Reader’s anguished cry, pure emotion. It’s even more beautiful how the instruments exit one at a time at the end, leaving nothing but her breaths as the final sound
First thing I thought of when I saw this category was Eminence Front by The Who:
TIghten Up by Archie Bell and the Drells
“Stairway to Heaven.” Acoustic guitar, recorder, vocal, electric guitar, bass and drums.
Nazareth - Woke up this morning. Not just only one instrument to start - but only one channel!
Just Like Heaven?
It kind of starts with drum and bass together, but technically you have the drums playing on their own to open the song with a fill before the bass comes in for four bars, guitar 1 for four bars, guitar 2 strumming for 4, synth strings for 4, lead guitar with counter melody for 16, voice comes in. Synths fade out, then back in towards end of verse, little piano is added, and then by the second verse, piano is fully present, providing countermelody.
First one I thought of as well. Speaking of the voice naming all the instruments, the expanded 2009 version of Tubular Bells features that voice, one Vivian Stanshall, providing a drunken extended tour of the mansion they are recording in, set to the sailor’s hornpipe. It’s an amusing addendum.
Maurice Ravel, Bolero – good for flash mob musicians. (It starts about 3 minutes in.) I was waiting for the guy with the surfboard chime in, but…
The Hollies “Long Cool Woman”. The intro is classic R&R construction, which is why it’s one of my favorite tunes.
Herb Alpert and the Tijuana Brass, “A Taste of Honey”
Bass drum, trombone, snare drums, trumpets, and the rest of the rhythm section.
Viv Stanshall, the same voice as on the Bonzo’s Intro and Outro mentioned above. Alan Rickman was the MC for Tubular Bells II and John Cleese did it for the remake. I have a single or a bootleg of Viv doing the TB 2 version also, but it wasn’t used.
If you’ll allow a liberal interpretation of “instrument” to include “a chunk of an orchestra”, then I reckon The Propellerheads do that 4 times in OHMSS: 0.15, 4.50, 6.50, 8.15. (all -ish).
(People who understand music - please feel free to explain to me why I’m wrong).
j
Anything by the Village Stompers. 1960’s fans will remember “Washington Square” - here’s another one done in the same style: Village Stompers - From Russia With Love - YouTube
“Memphis Soul Stew”
Marge Simpson’s version:
Hell’s Bells by AC/DC starts with
A 2000 pound bronze bell
Lead Guitar
Cymbal
Rhythm Guitar
Drums
Bass
Vocals
Essentially the first half(?) of Stop Making Sense, the Talking Heads concert movie, is like that. They even show rolling on the drum kit etc.
Granted it seems to be introducing new instruments with each new song. Still a lot of fun.