I could write a book on Advent Rising bugs, (getting stuck inside railing, jumping upward repeatedly forever, game triggers never activating to move the story on, etc.)
In Mario 64 I found one of the stars accidently on the second area by going into first-person view while standing up against a wall, which let you look through the wall, and all the others from that vantage point.
There's a PSII glitch where you can get two Neislashers, (or any other equipment) by equipping, removing and immediately exiting the equip menu. When you go back, it'll be equipped on Anna, but it will be available to even give to someone else with it still usable by her! I don't ever use it though, since it is such obvious cheating!
One of the Maddens, (2004, I think) had all sorts of bugs. Once there was a fumble when a QB was sacked an injured, and no one was able to pick it up, but it could be kicked. A 15 minute rugby session ensued until the ball was finally kicked out of bounds. Every once in a while you would score and get the ball back again at kickoff.
I'm sure there are many others, but those are the few I can remember right now.