The term 'independent movie' doesn't mean anything at this point really, and is certainly not a particular genre. Most of the movies on this list aren't particularly lowbudget or made anonymously. I've tried to stay off from obvious and egregious examples of fake indies though. I've chosen to include films made outside the established studio system, films made by little-known directors or films made with scarce means.
Some of these movies are just unconventional when it comes to its content and attracted a small audience when they debuted. With some exceptions i've excluded non-english speaking movies and movies from the UK and Australia. I've also tried to exclude most genre movies, which often are made with smaller budgets. I would also say that most of the films on this list are flawed, some are even bad but i would still recommend them.
Glaring omissions from Letterboxd.com:
The Tic Code (1999, Gary Winick)