A little selection
Movies:
Ninja Scroll: a masterpiece. Great classic plot and a brilliant and surprisingly rare mix of medieval japan with a zest of dark fantasy.
Perfect Blue: Hitchcockian thriller in the not so rosey land of idols.
Mind Game: a wacky mix of styles and an unpredictable story that branches out, loops and stutter for the most bizarre relationship and self-discovering dramedy.
5 cm per second: Youth, teenage years and adulthood of a man separated from the one he loves. Three brilliant short films rolled into one poignant feature.
Patlabor 2: Patlabor is a very original manga because it's about the everyday life of a police force in a japan where mechs are normal. It spawned a good TV serie and a very good first film, but Oshii pushes the originality a step further with Patlabor 2 as it suddenly becomes a reflection on war in the midst of a politico-industrial conspiracy plot.
The Castle of Cagliostro: Action, comedy, romance: the whole entertainment package is in the adventures of master thief Lupin the 3rd, delivered with infectious enthusiasm and energy by a young Miyazaki.
Macross Plus: The Movie: I guess you can watch that or the mini-serie. A tale about love, friendship turned sour, the weight of the past and daring to live the dream of our youth... all that masacrading as a brilliant fighter jet / mecha anime.
Also: Princess Mononoke, Nausica? of the Valley of the Wind, Venus Wars, Sword of the Stranger, From Up on Poppy Hill, Memories, Summer Wars, Grave of the Fireflies, The Sky Crawlers, My Neighbors the Yamada, Tokyo Godfathers...
TV Series:
Eden of the East: brilliant show that mixes thriller, comedy, slight anticipation and the most ambitious sociological themes in a very well produced package. It takes balls of steel to dare tell a story so over the top, ridiculous and barely intelligible, yet root it deeply in the realities of our time and go at it without an ounce of irony. 11 episodes and two movies.
The Melancholy of Haruhi Suzumiya: with the most ludicrous pitch (a schoolgirl overflowing with enthusiasm has, without her knowledge, the ability to shape the very fabric of the universe with her imagination), this anime offers the most experimental storytelling to ever hit mainstream TV along one of the weirdest brand of humor. Brilliant in so many ways. (Two seasons for a total of 28 episodes, but the episodes of the 2nd season are supposed to go between some of the episodes of the first and each episode has both a broadcast number and a chronological number... so yeah... I am not sure to know what is the best viewing experience).
Honey & Clover: a romantic dramedy with some coming of age elements as the characters are slightly older than in most shows (students at an art school, not high-schoolers). Many great characters and some absolutely brilliant sub plots that I often think about. 36 episodes.
Azumanga Daiyoh: the absurd, hilarious and bitter-sweet life of a handful of high-school students. The quality of the character portraits is that much more astonishing when you realize the source material is a collection of comic strips!
Serial Experiment Lain: it may have aged poorly, but this show was really groundbreaking at the time with its inclusion of internet technology, its plot based around the circulation of information and rumors and its oppressing atmosphere that makes the whole thing feels like a fever dream.
Also: Space Cobra, Patlabor, Hikaru no Go, Great Teacher Onizuka, Monster, Planetes, Moonlight Mille (these last five I have not seen, but I have read the original manga so I'll assume the TV shows can only be good with such brilliant source material), Toradora (superbly crafted 'opposites attract' dramedy in high school), Parano?a Agent, Maison Hikoku (the reference for all young adult romantic comedies), Kaiba (very original design for this very imaginative sf show)...
Manga
Do you also want japanese comic book recommendations? Because I have some too if need be!