Immediately I thought Fallout 2, because I've finished it so many times in every possible way, with everything in game done, and now when I play it I just skip dialog most of the time and just enjoy big shoot outs or trying weird things or just exploring and trying to find the secret encounters (even though I know them all now...). Yeah, it's just fun to play. I metagame the crap out of it though.
Pokemon Yellow, don't really play that for the story anymore, it was hardly revolutionary first play through (not that I clearly remember it all those years ago).
Lessee, Dark Crusade and Soul Storm I definitely don't pay much attention to the campaign story anymore, and usually play Skirmish or online anyway. Wizardry 8, played to death, am actually tired of 90% of it's story and only pay attention to the bits I don't do often or make me smile. Metagame that too. Unreal Tournament because I mean, come on, does anyone care? I make my own stories in Civ 4! Oblivion, I definitely only choose to care about certain bits of Oblivions story, generally just wing it and make up my own goals. I like the Dark Brotherhood and the Thieves guild, but unless I'm just absorbed in a little immersion roleplay, I don't care and I once again metagame the crap out of it. As well as just wander and as I said, make up things to do. I have played Chrono Trigger just to see different stuff happen, so I didn't really pay much attention to the stuff happening during game I'd see before. Fallout 3 gets similar, make your own goals and mess around gameplay, where not only does finding my dad become oddly unconcerning to me, but even doing anything remotely logical becomes alien.
Yeah, I game without concern for the story a lot. I mean when I'm playing Street Fighter I'm not always thinking about what the two characters backstories are and why they'd just fought the same match 30 times.