Weirdest one I can think of is the 'Self-Duplication Glitch' in UT2004, which as far as I know hasn't been fixed, so you can try it yourself if you actually still have that game. Get a Redeemer, launch it in guided mode (altfire), pause the game and become a spectator, then join the game again. Go back to where you were standing and you'll find an exact duplicate of yourself (which will just stand there). You can even kill the clone and the game will think you killed yourself. Telefrag yourself with the Translocator for a really odd death message: "[You] tried to go where no man has gone before." It's really bizarre.
Zelda: Link to the Past has a couple really strange ones (in the GBA version, at least).
The first one makes your boomerang go completely wacko, flying all the way across the screen repeatedly (ignoring Link for the most part) chopping stuff down and glitchifying the landscape (things get especially weird if you dig frantically with the Shovel while it's doing this). Getting it to do this is tricky though; you have to throw the Level 2 Boomerang, then quickly dash far away from it with Pegasus Boots. I usually do that by going to the lower right area of the castle or village, tossing the boomerang up and to the right, and dashing away to the left until I hit something. Once you have it going, it's extremely strange. And if you wait long enough it eventually stops being strange and returns to your hand on its own, almost as if this isn't really a glitch at all.
Also, if you use the Flute, dash up a staircase with Pegasus Boots, and press B to cancel, you'll be put on "turbo," able to walk at Pegasus Boots dashing speed. Be careful not to bump into anything, though, or you may end up bouncing back and forth all the way to the top of the screen, then looping back to the bottom and going up again, and so on infinitely.
This is what happens when you don't test a game thoroughly. But who cares? Glitches are fun!