Ever not quit a terrible or mediocre game? The bad game I didn't quit on was Klingon Academy. I'd spent money on it and it was Star Trek so I refused to give up even though it was awful. And I played it all the way to the end. And the only fun I had was seeing Chang get his eye plucked out in the video at the very start. Of course, I haven't got the patience for that kind of stuff now. I can barely bring myself to finish good games.
Generally I avoid terrible games...unless they're on the NES. Seems to me I can't find a game on the NES that I can't love. Conan (NES) was probably the closest to what you describe, but I enjoyed it despite it's (major) flaws, so I don't know if it counts.
Star Trek: Bridge Commander was pretty goddamn bad. The voice acting in particular made me want to vomit with just about every line. Forced my way through that, but still had to hit the mute key every time a character started talking.
Games people told me were great so I kept forcing my way through them before I finally quit.... Half-Life 2 Bionic Commando Rearmed (hate it!!!!!!) Mirror Edge (although I finished story mode and like it more as I got through it) Those are the recent ones.
When Bridge Commander came out, I was a little kid, so I thought it was AWESOME EXPLODY FUN! Looking back... meh.
Final Fantasy 9. I hated every second of that game. But it's FF, so I couldn't live without completing it.
Awful games I forced my way through = none I don't think I've ever played through any games that I hated. That soccer game was kinda meh, so I only played that once in a while. There have also been a few Flash games I've kinda disliked but played through to see if they got any better (with varying results).
Yeah, when I was younger I played through a lot of shitty games... Back then, when you purchased a game from the store all you really had to go on was word of mouth from friends and the artwork on the box cover. There were no online reviews you could read before you spent your money...
Ikari Warrors 2. You get up to the final boss, and the game turns off the A,B,B,A revival code on you, you die, the end. Beat it with Game Cheatie, a lame congratulations screen. And A,B,B,A ain't even a cheat, you can't even PLAY that piece of shit without it, there's bullets everywhere. The graphics are such shit, you can't even see 'em! Shitty ugly glitchy graphics, shitty scratchy garbled voice simulation, not fun, a miserable piece of shit from start to finish. But, I payed 50 bucks for the thing, so I felt like I had to ride it to the end to get my money's worth. Eh, I got 400 bucks worth of fun out of Megaman 2, so that fixed that.