Lots of great covers listed already.
+1 Greg Allman, These Days, but Nico’s is my fave cover of the song.
+1 for all the Johnny Cash covers. The American Recordings are sensational. Let me turn it around and point out a great cover of one of J Cash’s songs: NRBQ, Get Rhythm.
Other favorites that come to mind:
Beyonce, I’d Rather Go Blind (Etta James)
Leon Russell, Youngblood (Coasters; Pomus, Leiber & Stoller)
Beatles, Bad Boy (Larry Williams)
Fairport Convention, Keep It with Mine (Bob Dylan)
Sid Vicious, My Way (Frank Sinatra; Revaux, Thibault & Francois)