Part of my Japanese degree involved a focus on classic Japanese cinema, so…
Seven Samurai is arguably one of the greatest films of all time, no hyperbole. I mean, Kurosawa is one of the greatest filmmakers and this is essentially his greatest film (yes, we can all argue all day and night about this!). The Magnificent Seven is literally a Western remake of it (and is also a good film in its own right) and it can be fun to watch them back to back.
Rashomon is essentially the inspiration for every film with an unreliable narrator, definitely highly recommended. It’ll also help you understand one of the best throwaway Simpsons jokes.
Yojimbo is another Kurosawa film that, to me, rounds out the classic trilogy of Kurosawa/Mifune films. It’s probably one of the earliest antihero films. Another major Western, A Fistful of Dollars by Sergio Leone and starring Clint Eastwood, is based on this one. That film is also worth watching.
Tokyo Drifter and Branded to Kill are absolutely classic Yakuza films from Seijun Suzuki. If you saw Tarentino’s Kill Bill and were wondering where the inspiration for the entire end sequence came from, much if not all of it is from Seijun Suzuki’s work. The scenes cycle through major primary colors to the point where the main character’s outfits even change going from one room to another to match them, for example. These films are style over substance, over-the-top Yakuza-hero worship.
If you want the other end of Yakuza film-making, I’m pretty sure Takeshi Kitano’s films Hanabi, Violent Cop, and Sonatine are all in the collection and on Filmstruck. I don’t know how to recommend just one of these, but I guess I’d pick Hanabi (might go by “Fireworks” in English).
Kwaidan is a season-appropriate horror film containing four classic Japanese folk tales, in particular known for its take on Yuki-onna.
I’m going overboard, so the last one is Yasujiro Ozu’s Tokyo Story. In the history of Japanese film, I think it’s probably fair to say Tokyo Story is considered a must-see in the history of Japanese cinema.
As for non-Japanese films…
Brazil is a cult classic Terry Gilliam film about a disturbing dystopian future where a typo can destroy someone’s life and contains one of the stranger roles played by Robert DeNiro. I don’t know if the Criterion version is the director’s cut or not, but it’s actually pretty interesting to watch both the theatrical American (“Love Conquers All”) version and the original director’s cut (European release) because the film was heavily censored for US release.
I went on too long and recommended too many. Oh well!