Most guilds raided Molten Core many, many times to gear up everyone that needed it, and back at 60, I spent many weekends plowing through the lava packs down there. I've also run Shadow Labs many, many times since BC dropped, and Steam Vaults and the Arcatraz have also seen me go through quite a few times. Nowadays, Karazhan is probably getting the most traffic, although the cooldown on it probably keeps people going through there as often as they'd like. And don't forget lowbie instances-- I've been through SM and Deadmines on every character at least a few times.
But for me, the instance I've been through the most times overall is definitely UBRS. There were days, when I first hit 60, when we would run it twice a day. I've tanked it, healed it, and DPSed it. I've learned it, taught it, marked it, master looted it, done every quest in there and seen every drop. I've seen a hunter pull Drakk, a Mage pull Drakk, a Warrior tank them all, and even seen a Rogue do the pull. I can't count the number of times I've seen Eviscerate IX drop, and at one point, I could probably list off both all the pulls and Rend and Nef's dialogue to you-- without being near a computer. UBRS is my instance, and I'd put my runs through there at almost 100.
So what's the instance you've run the most, and how many times have you been through?