Paul McCartney is still alive.
Mike Watt - The Minutemen, firehose, Mike Watt
Jack Bruce-Cream
Flea, the Red Hot Chilli Peppers
Les Claypool of Primus, Sausage, and other miscellaneous groups.
Bootsy! 
Victor Wooten of Bela Fleck and the Flecktones.
Stuart Hamm
John Myung (Dream Theater)
Geddy Lee (Rush)
Stanley Clarke
Jeff Berlin
and personal, but fairly obscure, favorite: Brad Pedinoff from Harm Farm.
Paul is a fine singer and has penned some excellent tunes on occasion, but he’s not the best living bass player. If Carol Kaye is still alive, and I think she is, she would probably get my vote.
It all depends on what you mean by “best,” of course. You want most creative? Or the one who can play the most notes in a row without mistakes? Or the one who can play the most notes in a measure? (My computer wins both those easily.) Best bass soloist or best player in a supporting role?
Stefan Lessard, Victor Wooten and Bakithi Khumalo all have much to recommend them.
John Paul Jones, Led Zeppelin
Ryknow from Mudvayne is the best bass player to come out of the woodworks in the last 10 years. I won’t claim that he is the greatest of all time, but he can, and does, play the shit out of his bass. Most people probably write him off because he is in a nu-metal group, but they are really doing themselves a disservice by not giving him a good listen. He is a classically trained jazz bassist.
Victor Wooten and Stu Hamm (as already mentioned) are right up there at the top. Sir Paul is nowhere near their league.
I’m actually listening to Bela Fleck & the Flecktones Greatest Hits right now- Wooten is just brilliant. Les Claypool is also one of my favorites, he gets bonus points for being so innovative and progressive (especially nowadays). He’s also a great band leader.
Besides those two there’s Mike Gordon (of Phish fame), Phil Lesh (Grateful Dead), and Chris Wood (Medeski, Martin, & Wood).
…though Chris Wood mostly plays a String Bass (Double Bass), that probably wouldn’t count as bass guitar.
Otherwise, I would have nominated Danny Thompson. Who none of you have probably ever heard of.
Think Pentangle.
A few names I’d add: Oteil Burbridge and Alphonso Johnson. I’m a fan of Billy Cox (former Band of Gypsys) as well, though I’m told he doesn’t really play anymore.
Under the ‘can do things no one else can’ ruling I’ll back up Geddy Lee. The man is astonishing either holding rhythm or on runs. It’s just damn-all scary.
No knock on Geddy, whom I’ve long admired, but Geddy himself usually cites Jeff Berlin and Percy Jones as the best bass players around.
Personally, I’d LIKE to pick Paul McCartney, who always seemed to know just the write notes to play on every Beatle track… but I’ll nominate Tony Levin, both for his outstanding work with Peter Gabriel and King Crimson and for the quality session work he’s done with everyone from Pink Floyd to John Lennon to Paul Simon.
All this talk and no mention of Pino Palladino?