The most game breaking by far is name injection, being able to basically enter and run as much code as one wants by inputting strings into name/favourite/farmname/animal name boxes.
People often refer to item IDs, basically a small component of this, but there is so much you can do with name injection including completing certain checks, altering stats, changing when certain set events happen, etc. Especially in older versions where people have completed the entire community center in mere minutes abusing them.
On top of that, there are and were so so so many bugs in the game, most of which are useless but many of which can lead to some pretty fun runs, I'm someone who does a lot of bug hunting and I've found an absolute load when playing, usable or not. I'm actually using some components of name injection alongside a few other glitches in a video I'm making where I use it to advance a check the game makes to an earlier date, very interesting stuff when used in that way.