Usually bugs, glitches, and exploits degrade the experience of a game, but occasionally they can actually work to a game's benefit. In some cases, they can become significant enough to become part of a game's identity. In others, they make a broken game worth playing in the first place. Even without such legendary status, a given wrinkle in a game might simply make it more enjoyable or entertaining, or perhaps open up unexpected modes or paths of play.

What are some examples of these, and how did they improve their associated games?

I've found a fair few glitches that I find really interesting, and I'd be pretty interested in seeing some more pixel vomit too. For the ones I'll be listing, they are in old games but are very interesting.

Everyone's seen the MissingNo. glitch, but a far less famous (although in my opinion, more interesting) glitch is the Super Glitch. If you've got an emulator, I'd recommend doing a save state and messing around with it - it's results vary a lot.

A glitch available in quite a few games is arbitrary code execution. It's pretty interesting in general, as people can do loads of things, from loading up homebrew to replacing maps. Most of the time it is rather difficult or time consuming to do, but it's still fascinating to see.

This one technically isn't a glitch at all, but teleporting in SRS-based Tetris games is pretty cool. An actual glitch in Tetris (NES version) is pentrises don't clear all lines, which can make pretty interesting stuff happen.

EDIT: I nearly forgot about The Big Skip in Mario and Luigi: Superstar Saga. I recommend watching a speedrun from AGDQ, it only goes over it very briefly in the tasvideos page.

A lot of the time people will by default mark a game being buggy as a negative, and on the one hand I do understand this. But, at the same time, some of my most memorable gaming moments will be stuff like walking entering the wasteland to find an old barrel just spinning violently on the floor. Now, obviously a GAME BREAKING bug will always destroy an experience, but if you look at a civilian floating up a wall in spider-man PS4 and you're too worried about your immersion to find humour in it, I'm sure you're very fun at parties.