I like psychologically interesting movies, though my favorites aren't necessarily all like that. Some big ones for me:
Spirited Away - Miyazaki film about a girl who stops off at an abandoned amusement park while driving to a new house with her parents, who eat food meant for spirits and become pigs. She is unable to escape and ends up in the spirit world, enslaved by a witch who controls a spirit bathhouse as the protagonist attempts to rescue her parents and escape
Eternal Sunshine of the Spotless Mind - a man and woman break up in a very lightly sci fi world where it's possible to erase people's memories, and they choose to erase each other and inevitably the man regrets it as it's happening in his mind
Perfect Blue - animated Japanese film about an Idol who tries to escape Idol life by becoming an actress, which is an incredibly difficult and stressful process as she deals with a stalker and the film/TV industry at the same time. As the movie continues she becomes so stressed and tired that the lines between reality and the films she's in begin to blur together and she starts to lose grip on reality.
21 and 22 Jump Street - FUCK they're just well written and well executed comedy.