I’m defining band here as group that would probably be recognized by name by at least 60-70 percent of readers of the SDMB.
For example, I’ve read enough concert threads to know Jonathan Chance has seen Rush about 60 times in concert by now.
What band is in the lead for you?
I’ve seen The Who (including assorted solo gigs) around 30 times. Next runner up would probably be Todd Rundgren, who I’ve seen a dozen times or so.
I’ve seen the Moody Blues 5 times. They top my list.
Beyond that, I’ve seen Rush and Blue Oyster Cult 3 times.
I honestly don’t think there’s any band, by your definition , that I’ve seen more than 3 times in concert ((and, even with an expansive definition including any band, whether known or unknown, it’s probably not more than a dozen or so times.) Actually, I’m having difficulty thinking of any popular band I’ve even seen twice in concert. I’m guessing–although it’s outside my normal musical tastes–that Poi Dog Pondering is the most popular band I’ve seen most, I think about 5 times in concert, but I don’t think the name will be recognizable by 60-70% of SDMB readers. Beyond that, good question. I’ve seen a reasonable number of shows (probably about 100-150), but I tend not to repeat.
I think I’ve seen the Indigo Girls about 5 times. I like them, but not as much as that statistic would indicate. Just so happens that I’ve had multiple opportunities to see them live and the circumstances (location, cost, etc) all worked out.
I don’t go to many concerts.
Invoke my name and I appear.
Yes, I’ve seen Rush 54 times now, the first time being (I think) the Grace Under Pressure tour in 83? Maybe 84. Still, that’s a good 30 years of seeing them.
Others I’ve seen multiple times?
Cowboy Mouth, more than 20
U2 - 2
Mighty Mighty Bosstones - 3
Gin Blossoms - 3
Mary Prankster - 10 or so
Cranberries - 2
God, I don’t know. I’ve seen thousands of bands. Pretty much if a band came through DC in the late 80s through the 90s I saw them.
I saw The Kinks 6 times in the 1980s, and Ray Davies solo once in the 90s. So sevenish for them.
I’m assuming this is restricted to rock/pop bands and not orchestras. I’m not sure how many times I’ve seen the L.A. Philharmonic, but it’s in the double digits.
Cowboy Mouth (I’m the other Doper fan of theirs; they don’t fit the “probably be recognized by name by at least 60-70 percent of readers of the SDMB” criteria, but Jonathan Chance mentioned them and now I can’t leave them out!)
Paul McCartney
Billy Joel and Elton John (both separately and together)
I have the “U-Li-La-Lu” promo 12" that’s etched on the b-side. 
I also can’t think of any of seen more than twice, excepting Clapton and David Bromberg at three times each.
I guess I spread it all around. Some theaters I’ve been in 100 times, but always something new.
Reverend Horton Heat, Richard Thompson - 5 each I think
The Who, Jethro Tull - 3 each
Seen both ZZ Top and The Rolling Stones four times each, twice together.
Genesis 3x
Rush 3x
I’ve seen R.E.M. 6 times. 1985, 1989, two consecutive nights in 1995, 2004, and 2008. The 2004 show was at a small theater in St. Louis that was about 2/3 full, and it was BY FAR the best of all those shows.
And I’ve seen Rush 4 times, and each concert was in a different state. Des Moines in 1982, Omaha in 1990, and Kansas City, MO and Moline, IL several days apart in 1994. I considered seeing them in the Twin Cities a while back to keep this record going, but that fell through.
Moody Blues, Jethro Tull, James Taylor, Bruce Springsteen and Eric Clapton - all six times. That’s a pretty weird assortment. Clapton was with Cream, Derek and the Dominoes and solo.
Kiss - 5
REO Speedwagon - 2
Neil Young - 2
Hmmm…
4 times:
Sonic Youth, Throwing Muses, Mudhoney
3 times:
Jethro Tull, Juliana Hatfield, Primus, Sinead O’Connor, Alice in Chains
2 times:
Too many, at least 50 different bands.
Major acts:
I’ve seen the Allman Brothers Band four times: twice without the original lineup, once without Duane, and once without Berry.
Smaller groups:
The McKrells – 6 or so
Hair of the Dog – 7 or so.
I’ve seen They Might Be Giants 3x. Never seen any other band more than once.
Derek Trucks six times.
I’ve seen Wilco, or Jeff Tweedy solo, a total of 18 times, and that’s all since 2006.
I’ve seen Barenaked Ladies the most of any full band, at around 18 or 19 for the band and a few times for Steve Page solo.
Old 97s/Rhett Miller solo combined - probably a dozen or so times.