After reading the first page I was hoping to be the first to post Higher Ground, sadly you outposted me :( Higher Ground is one of my favourite Peppers' songs, and my favourite cover too. It's very different from the original, but well-executed, and you can still hear the original in it.
Some more covers I like:
http://www.youtube.com/watch?v=vXE3L0uxOU8
Mothership Reconnection (Daft Punk Remix) - Scott Grooves: cover from Parliament Funkadelic's Mothership Connection. I love electronic music, and this version (not really a cover, but whatever) keeps just enough of the funk to keep it true to the original.
http://www.youtube.com/watch?v=ZWSWdsIpLQk
Fire - Red Hot Chili Peppers: faster and heavier than Jimi Hendrix' original, but I like it like that. Not the best cover ever, but it's energetic and all, plus I love the Peppers.
http://www.youtube.com/watch?v=3P5Z606XhL4
Wild Thing/Purple Haze - Popa Chubby: excellent guitarist Popa Chubby covers excellent songs.
http://www.youtube.com/watch?v=bng3agUOYiI
All Along The Watchtower - Jimi Hendrix: the original is a bit dull if you ask me, and this adds the little spice it needs to be something good. Just noticed a lot of my favourite covers include Hendrix, hmm.
http://www.youtube.com/watch?v=qxl2epHHO8c
Snow (Hey Oh) - Igor Presnyakov: the original (Peppers again) is fantastic as well, but this is in my opinion the best cover of the song in existence. Presnyakov is a fantastic acoustic guitarist, and manages to play like a whole band with just his guitar. Most of his other covers are worth listening to as well, the man is all-round great and can turn even the crappiest songs into masterpieces.