Ivy Rorshach. She could crush Clapton, Hendrix and SRV under her spike heel in one fell swoop. Plus, she’s better looking than anyone else mentioned yet.
Yeah, that is some great geetar! But I’m gonna go with Greg Ginn.
Instead of repeating everyone else’s choices, I will offer some of my personal favorites (who surprisingly haven’t been mentioned yet):
Dick Dale, the king of the surf guitar
Brian Setzer (formerly of The Stray Cats, now on his own)
Brian May (of Queen)
Johnny Marr (of The Smiths)
And disruptive entity, I have to single you out and give you props for mentioning Poison Ivy Rorschach of The Cramps!
Tom Morello inspired many. 
How about bass guitar? Les Claypool plays a bass that sounds good by itself, which is something you don’t hear often.
Okay I will clarify my OP (sorry my mother has been in the hospital I havent been able to get on here). I mean the greatest overall no matter what genre or whether they play lead or backup. Just the absolute best of all time. Gadfly— White focuses on the music and his philosphy mirrors that. This trait seems all but lost in comtemporary music. Expano Mapcase---- thank you for the too young to know better but I am still a high schooler but dont let that make you think different of what i have to say. I call it as i see it. And thank you for all the players i have never heard of before. They are really letting me expand my musical horizons and helping us solve this argument. Keep em coming
With the new definition of “greatest” it is probably either Andres Segovia or Al DiMeola. Some others that come close are Steve Morse, Les Paul (He was amazing on a technical level and created the coolest electric guitar ever), Steve Vai, Satch and, believe it or not, Roy ClarK[#1]. And Chet. And…well I could go on for a bit.
Andres and Al are just amazing players. Technically they are just about perfect. I like both of them but they aren’t my personal favorites. I admire their technical abilities but other players, such as Randy Rhoads and Steve Morse are more to my taste.
Another amazing player I happen to know personally is Craig Small. He was (and still might be) on Shrapnel with a band called 9.0. Mr. Small rocks.
Slee
#1. Yes. Roy Clark rocks. I didn’t know this until I saw him live, by accident, at the New Mexico State Fair. I was simply awed by how good he was. I had no clue before that that he is an outstanding guitarist.
I’ll second Dick Dale. He pioneered the surf sound and staccato picking technique. His influence is ubiquitous, but he doesn’t get a lot of recognition.
And I’m going to nominate Junior Brown. He sings lead while playing lead, which is a feat you don’t see too often. And if that weren’t enough, he does it on two different types of guitars. He uses a “guit-steel” of his invention: six-string on top, slide guitar on bottom. It’s like watching a juggler. In the hands of a lesser musician it would be a gimmick, but for him it’s what he needs to do his music.
I’d have to throw a second vote in for Joe Satriani. I can’t believe that Eric Johnson’s name has not come up…
Mike Ness of Social Distortion has a poetic, sliding, lyrical quality to his solos that I think gets overlooked far too much.
Re: Harrison. I’ve always thought that his Beatles solos were the weak links in some otherwise very good songs. Can you point out some songs as references to his soloing ability?
I used to think Roy Buchanan was pretty fine. Is he still considered to be that good?
Karl, some people do think he’s that good. I know people who are nuts for Buchanan.
grisham, if you think Jack White is “Just the absolute best of all time,” you do need to educate yourself about music more. Sorry. He really is very good, but he’s not THAT good. If you think the best guitarist in history is a guy who’s been famous for all of 2 years, you do need to educate yourself about music more.
Start with Charlie Christian or Robert Johnson. Charlie played with Benny Goodman and many consider him the guy who defined the electric guitar solo as we know it. Robert Johnson (I guarantee you Jack White is a big fan of his, by the way) didn’t really solo, but he was a Delta blues player who wrote haunting lyrics and often sounded like he was 2 people playing at once. Both of these guys died in their 20s, and people are still imitating them.
I’m not a White Stripes fan, but Jack’s clearly a lover of the blues and I respect him for that (in addition to his playing). Check out some old blues guys.
My personal favorites not yet mentioned are Duane Allman and Derek Trucks.
Give Jack White another ten years in his career, and he might crack the top ten. He really is very good, but when you talk about the ‘best’ guitarist you are in some extremely rarified territory.
I recently bought a Stevie Ray Vaughan concert DVD, and his guitar playing is beyond fantastic - it’s almost supernatural.
For me, there are three things that make a great guitarist - innovation, emotion, and technical ability. Steve Vai and Ynvie Malmsteen don’t make my list of greatest guitarists because, while they both have gobs of technical ability they employ it in a way that to me isn’t particularly emotional or innovative. On the other hand, someone like Slash doesn’t have their lightning fingers, but he can drip emotion from a guitar. Stevie Ray had both emotion and technical ability.
So far, Jack White gets major points for innovation and emotion. People under-estimate what it takes to get that full sound from a band without a bass player. He’s a good slide player, too. But as a guitarist he still hasn’t done anything that made my jaw drop with its sheer brilliance. And I’m a big fan.
Marley, for the record, Robert Johnson did in fact record one guitar solo, on ‘Kind Hearted Woman Blues’ – the fact that his style focused on playing bass and rhythmn simultaneously made it difficult for him to play solos. (This assumes that we define ‘solo’ as lasting an entire verse – RJ of course played lots of short fills and turnarounds).
I was counting one solo in 29 recorded tracks as “didn’t really solo.” 
Assuming the recordings are a good sampling of his repertoire and style, I think it’s safe to say he didn’t do it much.
Yngwie Malmsteen, Olaf Lenk, Timo Tolkki, Steve Vai and Joe Satriani.
I am surprised no one mentioned the first three already. The Straight Dope must not listen to must power-metal (some of the most technically demanding guitar-play IMO).
One of my favorite artists of all! I love his solo work even more than his Social Distortion stuff, but I admit I never really thought of Ness as a stand-out guitarist. He is a wonderful soloist, now that you mention it. “Cheating At Solitaire” is definitely a “desert island album” for me!
Geez, nobody’s mentioned either Bo Diddley or Bill Frisell yet?
Another guitarist not mentioned so far is the late, great, Michael Hedges.
And you simply can’t forget the late Terry Kath of Chicago.
If you can get hold of a copy of Chicago II, listen to a track called Free Form Guitar. Nothing more than Kath, a Stratocaster, and a Fender amp.
Breathtaking guitar work.