Some good shouts in here. I'll add a few I don't think have been mentioned yet:
Make Way for Tomorrow (Leo McCarey, 1937). A classic, which inspired another great film, Tokyo Story.
Wendy and Lucy (Kelly Reichardt, 2008)
Burning (Lee Chang-dong, 2018)
Seconds (John Frankenheimer, 1966)
Kes (Ken Loach, 1969)
The Elephant Man (David Lynch,1980)
Mulholland Dr. (David Lynch, 2001)
Cold War (Pawel Pawilowski, 2018)
Stroszek (Werner Herzog, 1977)
Brief Encounter (David Lean, 1945)
Irrespective if it qualifies as a movie, the ending of Twin Peaks: The Return effected me for weeks. So deeply, chillingly unsettling.
Yeah, that final shot of the girl looking over the balcony, so powerful. Kore-eda films in general are quietly devastating: Nobody Knows (that pink suitcase), Maborosi, Still Walking, Like Father, Like Son. He imbues a very strong feeling of melancholy and the bittersweet into his humanist stories, and his films hit you hard without being overly dramatic about it.