He played Mobile a lot in the early days. The band my Dad managed would play the same bars. When I was in the Army over in Germany he had a sold out show. I commented to a friend that I was surprised that he sold out a show in Europe. My friend was like of course he sold out he’s huge. I never knew he was anything more than a southeastern act. He couldn’t be famous, he sang with my Dad’s friends in the back yard for crying out loud.
I have seen Pink Floyd every year, many times, since 1967 until the last Pulse concerts. I have no idea how many times that is - but a lot.
Crosby, Stills & Nash twice, Grateful Dead twice, Black Sabbath twice, Asleep at the Wheel twice. Looks like two performances is my limit.
Bob Mould / Sugar: in excess of 12 times, lost count.
Naked Raygun: 6+
Wilco / Jeff Tweedy: 6+
R.E.M.: 3
New Order: 3
fIREHOSE: 3
Morrissey: 3
Pegboy: 3
Poster Children: 2
Califone: 2
Shellac: 2
The Charlatans UK: 2
Seam: 2
The Connells: 2
Spiritualized: 2
Curve: 2
808 State: 2
Grant Hart: 2
Guided By Voices: 2
Sonic Youth: 2
Jane’s Addiction: 2
the Jesus Lizard: 2
Tar: 2
Matthew Sweet: 2
Sloan: 2
I’ve seen the Eagles once.
Other than that, I got nothing. 
I lost track of how many times I’ve seen Gogol Bordello now. But it’s got to be closing in on 10, I figure. Nothing else comes close, really.
Another Deadhead here, although not quite as avid as many. I think I saw them 14 times, and probably another 10 or 12 shows from side projects or post-Garcia configurations.
String Cheese Incident would be in second place, I think about 10 shows from them. I haven’t been to see either them or a Dead related band for 7 or 8 years, I’ve pretty much lost my patience for those kind of shows in big arenas.
I saw The Clash eight times altogether: first time at a tiny Plymouth club on the 1977 Out of Control tour, then two nights running at the New York Palladium (including the night Simonon smashed his bass for the cover of London Calling), then three nights running at London’s Lyceum in 1981, then at St Austell in Cornwall during the Combat Rock era and finally in London again with The Pogues in support. First gig was the best.
I’ve seen Bela Fleck around 20 times- a dozen or so with the Flecktones and several other times with the many side projects he does, such as playing jazz with Chick Corea, classical with the Colorado Symphony, African music with Toumani Diabate, Vusi Mahlasela, Bassekou Kouyate and others, whatever you call the stuff he does with Edgar Meyer, and even traditional bluegrass with the Telluride House Band.
In second place is Stevie Ray Vaughan; I saw him 11 times.
The bus came by and I got on—that’s when it all began. There was Cowboy Neal at the wheel to a few hundred Grateful Dead shows.
Was never able to figure out the exact number. A friend and I tried to once in our early days, back when the best we could count was up to fifty or so, but even then it was vague. Shows weren’t something to count at first; the band was just this, you know, amazing band to see. Again. And again. Like the journeys between songs, who knew where they were taking us?
Not having a starting point, we never put an effort into keeping track after that. Sure, we had innumerable stubs (with scrawled setlists on them), but the slightly chaotic nature of things meant many were lost. And things were fuzzy for various reasons. And given the scene, we were at more shows than we were in, so who can remember if we caught all three nights at the Garden, or just the opener? And as tape collections grew and intense listening followed so did analysis. A room full of people talking about the Darkstar tease or what Bobby was doing here or there kind of blurred the lines. Plus you didn’t always tour with the same friends. So later attempts here and there never amounted to much more than ballparks.
Caught an unknown, large (but not as absurd) number of shows after Jerry left us. It’s funny that it’s been almost twenty years and I can still get choked up. 
There are a lot of close runners-up. Caught countless Allman shows. We catch P-Funk three or four times per year. I like posting a seasonal concert thread, so the number of shows has lowered but we keep on dancing.
Except for Gogol Bordello. Very jealous LM. We’ve had three or four separate sets of tickets—actual tickets, not just plans—but something always happens and we miss them. You’d think we’d be better at it by now.
Invocal. They’re a lesbian band and not that well-known, but they should be. I do know the band but have always gone to see them because I actually love the music. Otherwise I’d have hung out smoking while they played or something.
When/if I’m able to dance again they’re the band I most want to see.Not till then, though - they require jumping around.
Jack White in various incarnations - 17 times
(White Stripes 8 times, Raconteurs 6 times, Dead Weather once, Jack White twice)
Kasabian 8 times
Black Rebel Motorcycle Club 8 times
Hayseed Dixie 6 times
Pulp 5 times
Fun Lovin’ Criminals 5 times
The Fratellis 5 times
The Mission 5 times
Franz Ferdinand 5 times
The Datsuns 5 times
Muse 4 times
Roger Waters doing the Wall 3 times (2011 &2013), Pink Floyd doing the Wall once (1981)
REM 3 times
Robert Plant 3 times
Pendulum 3 times
Manic Street Preachers 3 times
Twice:
The Cult, The Pet Shop Boys, Bryan Ferry, Bruce Springsteen, Foo Fighters, Eminem, Ian Gillan, Motorhead, Jane’s Addiction, Wolfmother, Placebo, Faith No More, Jarvis Cocker, Budgie, Yes, Crowded House and probably others I can’t think of right now.
Scougs, we’ve never met, both live in the same medium-sized city and have both seen some 3-figure number of gigs each, many of them in Edinburgh I presume.
And the only overlap on your list is that we’ve both seen Yes! Probably not at the same gig, though; I saw them once in 1974 at the Usher Hall.
(I assume there will be some more overlap in the groups neither of us mentioned having seen though!)
Grateful Dead about 100 times.
Crosby, Stills and Nash about 20.
Yes about 12.
Santana about 12 too.
For the Gogol fans out there, it’s ever-so-slightly tangential music-wise, but are any of you fans of Balkan Beat Box? One of the most high-energy shows I’ve been through was BBB and Mucca Pazza doing a free show at Lincoln Center.
Sample! War Again, Digital Monkey, Bulgarian Chicks
54:40 - 5 times.
The last time I saw them was at a small bar here in town. I was at the very front - so close I had to actually lean back so the bass player wouldn’t hit me in the head. It was awesome.
I’ve played bass in a lot of small bars, but I was never tempted to hit anyone in the head no matter how close they were to the stage!
Coheed and Cambria. 7 times.
Motorhead. 11th show coming up in a month.
2x Slayer, 2x The Dubliners, and I don’t think I’ve seen any other band more than once that is well-known by OP’s criteria.
I’ve never gone to a concert in my life.