Mick’s voice? (Just a guess).
I think the same thing is true of Ray Davies and The Kinks.
Those voices are a “signature”, I think.
Quasi
Social Distortion has covered “Under My Thumb” twice, once on Mommy’s Little Monster and once on White Light, White Heat, White Trash. Both versions are great. They also covered “Backstreet Girl” on the Prison Bound album.
The London Symphony Orchestra’s “Sympathy for the Devil” was pretty good.
What?
Jane’s Addiction did a pretty good cover of Sympathy for the Devil.
SftD is one of the three Stones songs I actually like (yes, that one too: no, not that one, not that other one either.)* It’s idiosyncratic enough that despite it not seeming to work and being one big directionless jumble, I like it anyway. No one else’s attempts can measure up.
*Paint it Black, and Girl, I Miss You.
To further amplify (!) what I posted above about the “signature” voices:
When Ray Charles (himself a “signature voice”) covered The Beatles’ Yesterday, it was well done and I accepted it, but when Aerosmith covered Come Together, I thought “Meh, still prefer the Fabs.”
Didn’t mean to veer away from the OP, just wanted to add this.
Another example would be Ray’s version of Georgia, later covered by Willie Nelson, another “signature voice”.
Okay, I’ll shut up now. 
Q
Weird Al did a polka medley that was just Stones songs. Of course Weird Al has done a polka medley of just about everything, but its good for a giggle or two. Hearing Weird Al sing “Laughter joy & loneliness & sex & sex & sex & sex!” is amusing.
Inkubus Sukkubus does a nice “Paint it Black”.
The Who released “The Last Time”/“Under My Thumb” as a single in 1967 to show support for the recently-busted Stones.
ah, Under My Thumb* – there’s a version of that on the Steve Winwood Greatest Hits set – too lazy to look for exactly who did it – it was circa Blind Faith.
Guns N’ Roses did a pretty good cover of Sympathy for the Devil
As did Blood Sweat and Tears. Worked it into Symphony/Sympathy For The Devil with movements. It’s on BST&T 3.
Quas
Just got this – pretty damn entertaining. I spent my commute this morning meditating on Mick and Keef as songwriters, which, dammit, it turns out they are. Who knew?
Thanks for the recommendation, Quasi!
Going over my list of MP3s, I have Steve Earle and the Cowboy Junkies both doing their own covers of “Dead Flowers.”
I don’t have this one but back in the early 80’s, Kim Carnes (of all people) did a gender-switched version of “Under My Thumb.”
Second for Linda Ronstadt’s version of Tumblin’ Dice
Jonathan Round [I think is the name of the artist] does a positively demonic version of Sympathy for the Devil. Has anyone outside of St Louis ever heard of it? The song is played occasionally on the K-SHE Klassics show on Sunday mornings.
When I was about 15 years old I thought Grand Funk Railroad’s cover of Gimme Shelter was much better than the original. I am now older and wiser, Gimme Shelter is my favorite Stones song now.
Tori Amos does a good cover of “Angie.” I don’t know that it’s on any album of hers; she does a lot of live covers.