I am not referring to "easter eggs" (or hidden moves) but rather using a normal move in a strange way.
For example, in a baseball game from EA you can hit the batter when you pitch. If you are bored you can keep beaning the entire opposition until you have completely amused yourself. They have a bunch of different beaning moves so you can keep experimenting.
In Virtual Fighter 3 for Dreamcast either character can leave the ring. If you knock your opponent out (a win) you can then jump out without penalty (you have to do it quick). If you aim correctly, you can land on him and bounce him across the ground. Also, off some of the buildings your guy goes into a swan dive and yells right before.
I have also seen strangeness like in Bass Fishing, sometimes when you are reeling in a fish, it says "scene missing" for a second and then returns to the regular action. Also sometimes the fish bends in a strange way or goes through a solid pole.
Even mini-golf is weird because if you are playing an opponent both of you can have a ball occupy the exact same spot or one ball can pass right through another.
Feel free to add any other weirdness.