please change thread title to 'Incredibly Obvious Anime Movie Suggestions' :D
all good suggestions so far to be fair, but for weirdness:
Angel's Egg - ultra mysterious and glacially paced symbolist/surrealist/dreamscape by Mamoru Oshii (best known for the amittedly great Ghost in the Shell but director of many much weirder and more interesting anime and live action movies. Ghost on the Shell 2 is visually astonishing and mind-melting, I much prefer it to the original. Some of you military fans might like the aesthetic of his live action 'Avalon', shot in polish in poland (in order to be able to hire soviet tanks etc) and kind of a 'no THIS is how it's done' to the wachowski's, kind of like a Svankmajer/Tarkovsky-styled version of The Matrix!?) and his anime Jin-Roh: The Wolf Brigade set in an alternate 1950s Germany-occupied Japan
Night on the Galactic Railroad (1985) - somewhat similar masterpiece of surreal depressing slow beautifulness, based on the work of author Kenji Miyazawa. There's a new kind-of sequel just recently released, based on one of his other works, I haven;t seen it yet but looks promising:
http://www.pelleas.net/aniTOP/index.php/the-life-of-gusko-budoriGenocyber - 90's Manga Video guilty pleasure, ultra gorey weird trash. Children graphically machine-gunned to bits, incongruous low budget 'experimental' moments, less interesting and gorey last episodes.
MIND GAME - astonishing feature, amazing styles/animatio/direction/story/everything. All Masaaki Yuasa's other films/series are great too: Kemonozume (more shocking/weird/violent), Kaiba (looks mega cute but just heart-rendingly tragic and brutal story), Tatami Galaxy, etc etc etc
Serial Experiments Lain - post Eva/X-files weird cyberpunk series, made at that beautiful point where everything was about computers/virtual worlds but mostly still analog in practice so a nice mix of styles in a kind of crude way. Content-wise really great and creepy, the ending is pretty emotionally brutal.
The current series 'Flowers of Evil' (Aku no Hana) seems interesting, perverse and literary (based around Baudelaire's 'Les Fleurs du mal'), with unusual rotoscoped animation style:
http://en.wikipedia.org/wiki/The_Flowers_of_Evil_%28manga%29#AnimeNeon Genesis Evangelion - greatest artwork of the 20th century. Does anything any power electronics ever did (x 9000) and more while being the #1 most popular thing in Japan at the time. (More hyperbole some time maybe, or something more sensible to actually entice you to watch the thing)
...Uhh i'd better do some work. More recommendations soon.
A good site for finding interesting/obscure/animator-driven anime (and hardcore info about the actual animation + direction of these things which is very rare amongst the 10000s of superfical immature anime fandom sites online) -
http://www.pelleas.net/aniTOP/index.phpre: "weird, offensive, messed up" the 'mainstream' of modern otakuism consists almost entirely of sadeian/lolita/misogynist Moé creepyness but it's not not in a 'grim' 'evil' style so all you creeps probably wouldn't even like it. ^__^