Alain Resnais' 1961 French film Last Year at Marienbad is both highly regarded and almost entirely inscrutable. Anyone who claims to understand it is lying: Last Year at Marienbad is not a film you "get," but a film you have theories about. (If you Google it, you turn up essays like "Last Year at Marienbad: An Intertextual Meditation.")
This isn't to say it's boring. Sacha Vierny's cinematography is flat-out stunning, the pacing and repetition of scenes and dialogue completely hypnotic, and as much as you know the pieces are never going to fit together, it's impossible not to keep shuffling them this way and that, trying to make sense of it all.
The plot: A man (called "X") approaches a woman ("A"), claiming they met the previous year (at Marienbad, natch), and that she promised then to run away with him. A, though, is married and doesn't seem to remember X at all. A and X enact the same scene again and again, in the corridors and gardens of the opulent hotel in which the film is set. Mirrors are important. Games are important. The hotel is full of simulacra, yet reality itself is utterly indefinable. Memory merges with reality merges with dreams. I didn't understand it at all, but I'd watch it again in a heartbeat.