The Stones do a fine version of Fortune Teller.
Did they cover any other songs?
Going To A Go Go.
Originally done by The Miracles before they were Smokey Robinson and The Miracles even though Smokey was in The Miracles.
It’s All Over Now (Bobby Womack)
Gloria (Van Morrison)
Like A Rolling Stone (Bob Dylan)
Not Fade Away (Buddy Holly, though I don’t know if he co-wrote it with someone else)
Love in Vain – Robert Johnson
“Hey Mona” - Bo Diddley
Just My Imagination (originally by The Temptations)
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I Can’t Be Satisfied** (Muddy Waters)
http://www.coversproject.com/artist/rolling%20stones/
The first section, bottom part is people who have covered them.
They did Route 66.
“Shake Your Hips” by Slim Harpo
Lots, especially in their earlier days. The Stones first album only contains one Jagger-Richard song, and one Nanker-Phelge (i.e., group composition). The other 10 songs are all covers.
Five of their first 6 singles were covers or written for them (I Wanna be Your Man written for them by Lennon & McCartney), and the only one that was a Jagger-Richard song was not even released as a single in the U.K. (Tell Me - which is also the one Jagger-Richard song on that first British album - great song though.)
The early Stones were basically a cover song band.
Here is the Stones discography on Wiki. Go through it, and, especially in the '60s, you will find lots and lots of covers.
**njtt **offers a nice summary. If you are looking to get into the Stones - and I would strongly recommend it - then, yeah, please understand that they went through at least 3 distincts phases -
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Early - started as a cover band; they didn’t know how to write songs and only started when their manager grabbed friends Keith and Mick and literally locked them in a room.
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Middle - begun songwriting with 60’s pop songs - 19th Nervous Breakdown, Paint it, Black, Satisfaction, etc. Some covers, but really turned over to their own stuff.
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Golden and beyond - from Let it Bleed and Beggars Banquent on, when Keith started writing songs in Open G tuning and moving the anchor from 60’s pop to older blues sources then their songwriting got more sophisticated and their covers became much more thematic and in their own voice. Stuff like Shake Your Hips off Exile just fit perfectly in their new approach to the music…
Two Nanker-Phelges, actually, although “Little by Little” is such a blatant rip-off of Jimmy Reed’s “Shame Shame Shame” that it might as well be a cover.
“The Harlem Shuffle” (Bob and Earl).
They did a cool version of Muddy Water’s “Champagne And Reefer” on the Shine A Light concert film, with Buddy Guy joining in to boot.
“Just My Imagination” has been mentioned, but they did several Temptations covers. They’ve done their own verisons of “My Girl” and “Ain’t Too Proud to Beg.”
Supposedly the very first song the BBC played in FM stereo.
“Ain’t Too Proud to Beg”
Also Stop Breaking Down. They didn’t credit Johnson on the first release of the album, and I saw Taj Mahal sing a song criticizing them for ripping off blues songs in general just after.
I’m A King Bee - Slim Harpo