I think my favorite glitch was when I was playing Grand Theft Auto III on the PS2.
Looking at my handy, dandy cheat sheet (as you cannot play the game WITHOUT cheats), I decided on the tank an the good old flying car cheat. As most people new at that time, by turning your barrel backwards on the tank and using the flying car you'd be able to get much higher hangtime. Since there weren't any ramps (at least not that I remember), I decided to launch myself of the median of the bridge connecting the two parts of Liberty City. When I did this, however, I kept on firing the tank's artillery and kept going upward...and upward and upward. Even when I put the controller down, it went kept firing off artillery off screen presumbly somewhere in the sky going through the game's boundaries with the text at the bottom reading "Now leaving Liberty City" The camera angle focused at the sky with a bit of building thrown in and the tank was outside of the physical realm at this point.
I'm sorry I can't find a video of this, but I doubt it happened very often, if at all to anyone other than myself.