Who sang back-up vocals (uncredited) on "You're So Vain" by Carly Simon?

Mick Jagger.

https://en.wikipedia.org/wiki/You%27re_So_Vain

DaMoopies' answer is correct, but I have to share this: when Carly Simon performed "You're So Vain" on season 1 of "Saturday Night Live" back in 1976, cast member Chevy Chase (who was already a recording artist as drummer and vocalist for the early-1970s band Chameleon Church) impersonated Mick Jagger on backing vocals and cowbell (note the Jagger-esque scarf that he's wearing), as the linked video shows. The story (which I can't verify) was that Jagger had said he'd appear but then had a conflict:
https://archive.org/details/carly_simon-youre_so_vain-snl_1976