"The Last Time" is officially credited as the first UK single by the Stones written by Jagger-Richards. The song was recorded in Hollywood, CA in January 1965 and went to no. 1 on the UK charts just a few months later.
Keith Richards much later admitted however, that "The Last Time" was lifted from a 1955 Staples Singers song called "This May Be The Last Time".
Here is the Stones version:
And the Staple Singers:
Larceny notwithstanding, Brian Jones' guitar riff in the Stones' version is one of the most distinctive of its era and a great moment in Stones history. It's a terrific single, one of my favorites.
And almost 50 years later, the Stones are still playing this song. Here's a recent live version from a 2013 show in Los Angeles:
"The Last Time" has been covered by many, including Bruce Springsteen, Arcade Fire, The Who and Bobby Bare (Sr.). But the band most associated with the song, other than the Stones, is the Grateful Dead who frequently covered it, including this slow building, excellent version from 1992: