Missingno is one of my favourites. One slight oversight causes a huge chain reaction that can just break everything. So much just happens to work out perfectly.
Integer overflows tend to be fun too. In Super Smash Bros Melee you can suck so bad at the first level you get a negative score. The resulting overflow gives you 999,999,999 points. If you do this with every character though, your total score also overflows and becomes negative.
Lufia's level zero glitch is pretty crazy just for how easy it is to trigger and how bizarrely and thoroughly it corrupts the game.
Older PS2s apparently had a similar glitch, where just pressing a couple buttons at the right time glitched up the menu. This let you bypass the region restrictions on DVDs.
Link to the Past had a nice glitch where saving and quitting in midair allowed you to knock yourself under the ground. The GBA version kept this glitch, and you could walk right out of reality. When you do so, the game briefly freezes, and then BOOM, several different sound effects at once, EVERYTHING is corrupted. It's pretty nuts.
Mega Man 64 has a little feature where doing "evil" things turns your suit darker and doing "good" things turns it brighter. You can essentially rob a bank at one point, and it just goes black. During the endgame sequence my brother recycled a can laying on the ground, and instantly turned black.

Who knew recycling was so terrible?
Meanwhile, in one place you can actually
punt puppies and nothing happens...
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