As a busy student who has little time and money for games, I find myself unfortunately only able to purchase about 2 games a year (if that). This means that on the rare occasions I do buy games, I tend to look for the GOTY Editions since they give me the most content. However, this also means that I hold back on purchasing games, thinking that GOTY Editions will release when in fact they might never. So my question is how are GOTY Editions brought about, and how long should I wait until I give up on waiting for the never-arriving GOTY Edition and just buy the darn game? Thanks in advance!

Game of the year editions are born when publishers decide they want a new $60 product on store shelves. There's no real way of predicting it, other than looking at a publisher's past history. If it's in a game's plans to ship a bunch of DLC, it's almost certain to get a game of the year version these days.

If a piece of single-player dlc for a game comes out, say, within 3 months of the original game, then the GOTY edition will usually come out about 1-2 years after the original game is released (except for EA Games).
If dlc comes out ages after the original game (Bioshock Infinite for example), then the GOTY edition will probs come out after you've graduated anyway so you might as well get the damn game, and by the time the dlc comes out you'll have enough money for that by then.

If you're on a budget, Game of the Year Editions are rarely a good bet. They're generally a way for publishers to keep charging $60 for the same game, by bundling it with some middling DLC that you probably don't want/need. You're probably better off just waiting a while after the core game is released, and picking it up on sale or used. It'll be cheaper than paying full price for GOTY editions.

Yes and yes. GOTYs are for when the game has finally hit the bargain bin and the publishers are hoping some extra swag will convince consumers to pay full price. Sometimes they're worth it (Fallout 3, Borderlands 1) sometimes they're not (Saints Row the Third, Arkham City.) Your best bet is to wait around until the GOTYs themselves hit the bargain bin. Of course by then the original game is years old...