Aerosmith doing “I’m Down”.
Tragic
They didn’t do many covers but maybe Roll Over Beethoven.
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Not sure this one counts, but has to be included for it’s sheer brilliance…
I have always thought in the back of my mind Oasis were influenced by The Rutles more than The Beatles 
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Jeff Beck’s cover of She’s A Woman on his Blow By Blow album is quite good I think. On the whole, however, I prefer the originals.
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My choice too.
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Milos and Tori Amos. She’s Leaving Home.
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I really like this, I’m a Tori fan anyway and the lyrics are wonderful.
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Game over 
Jeff Beck the guitar god!!
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This one, I humbly submit, is just about as good as the original, given the great singer, the energy, and of course the dedication at the beginning:
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I can only think of one Beatles cover that is superior to the Fabs original, and that’s EW&F’s sizzlin’ version of “Got To Get You Into My Life”.
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Seeing someone mentioned Siouxsie’s cover of Dear Prudence reminded me that she also recorded a fine version of Helter Skelter.
For No One is a truly beautiful song, but it’s also a delight to hear Emmylou singing it.
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Will probably be controversial, but Michael Jackson’s cover of Come Together would be my nomination.
Mind you, I’ve never been the biggest fan of the Beatles, so take my suggestion in that context.
Mark
A couple spring to mind - personally not a great Beatles fan but anyhoo
Morgan James and Hayley Reinhart - Dear Prudence
Or if you want to go whole hog with Morgan James
The White Album
A Beatles cover better than original is a tough assignment. To my ears nobody’s ever managed it. The best of them are the ones that do it so differently that they almost create a new song. Joe Cocker’s ‘with a little help…’ is a prime example, helped by the fact that the Beatles own version is pretty naff by their standards and Joe’s version retrospectively helps show that George Martin’s production choices in the Sgt Pepper sessions undersold another great Lennon/McCartney composition.
For me, Jeff Beck’s ‘day in the life’ is worthwhile, but definitely doesn’t improve on the fabs.
The one I enjoy most ( and most frequently) is The Carpenters ‘ Ticket to Ride’.
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I agree that one is a belter!
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My first nomination would have been Joe Cocker - With a Little Help…
Rather than a cover a revision, a bit like Hendrix playing Star Spangled Banner.
I am sadly old enough to remember lunchtime programming on BBC Light Programme, where a “choir” and light music orchestra would murder that weeks top ten, cloned badly rather than covered.
So, my second choice, Richie Havens - Here Comes The Sun.
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Third Prudence by Siouxsie.