Ok, here are my recommendations with a bit more detail.
- Serial Experiments Lain: There is one word to describe this anime, and coincidentally it is the title of the first episode -- WEIRD. This short 13 episode anime is a trove of mind-numbing psychological craziness that will leave you hopelessly confused until the last episode...and maybe even beyond. An older show directed by the legendary late Ryutaro Nakamura, who was known as a horror director. He brings a unique take to the anime industry through this one-of-a-kind anime experience. "PRESENT DAY, PRESENT TIME", this show sheds unique and accurate light on the world of technology and "hacking".
- Kino No Tabi: While this show is similarly directed by Ryutaro Nakamura, it is quite different from Lain. It is a slower-paced, episodic anime that delves deep into the quandaries of life, existence, and reality. The quote that opens the show represents it the best -- "The world is not beautiful, therefore it is [beautiful]."
- Kaiba: Visionary Masaaki Yuasa brings his weirdness to a 12 episode series. A dystopian world where the rich can extend their lifespans by supplanting their memories into artificial bodies but the poor must suffer, Kaiba follows a young man with a hole in his chest as he tries to reclaim his forgotten memories. This show is quite the trip -- fantastically animated with a strange Yuasa tinge.
- Haibane Renmei: The Haibane are a group that wake up with no memories in a strange town with huge walls. They grow wings and have halos that float above their heads. They must learn to grow and live together with the human townsfolk as they try to learn the meaning of their existence. Existentialism at its finest. A bit slower, with a "slice of life" tone towards the beginning, this show brings some curious questions about the lives we live in general.
- Akagi: Protagonist Akagi is the very definition of a badass. Starting his journey as a 13 year old who wanders into a Yakuza Mahjong parlor, he ends up playing Mahjong with increasing stakes and increasing badassery. His life is always on the line; will he continue unflinching, or will he dip too deep into danger and lose it all? This show has some interesting character designs.
- Kaiji: By the same group as Akagi, Kaiji is somehow a much different show. Still about the underground Yakuza gambling world, the show follows young Kaiji who is a hopeless and pathetic loser in life. He is addicted to gambling and lives in constant fear. How will he survive when he gets unwittingly flung into the underground? This is a bit of a longer show -- two seasons that are around 20 episodes each.
- Texhnolyze: Another Yoshitoshi ABe show? Yes. Because ABe is a genius. It's hard to describe the futuristic dystopian world of Texhnolyze, where people can have their limbs replaced by machines and the underground city is ruled by gangs. Watch the world descend even further into death and darkness as it spirals out of control. Slow to start, but definitely worth it for a brilliant ending.
- Inferno Cop: Inferno Cop is a cop who has come back from Hell for revenge! About 12 episodes of 5 minute-long fun. Ridiculous Trigger humor that rarely misses delivers at its finest. Animated even worse than South Park, this show holds no punches when it comes to making fun of everything around it. Reads a little bit like Kung Fury.
- Space Patrol Luluco: WARNING -- watch Inferno Cop First! Trigger delivers again with 12 episodes of 5 minute-long fun. Join Luluco as she gets forced into joining the Space Patrol, a group that believes that JUSTICE NEVER DIES! Watch them take on wacky and crazy supervillains that have abducted the entire city of Ogikubo.
- Jin-Roh: Political Military Thriller. Who is the mysterious Wold Brigade? What is their role in the development of Post-War Japan? A calmer, more psychological military piece that ponders morality.
- Patema Inverted: Half of humanity has had their gravity reversed. Patema, being one of the inverted, accidentally wanders to the surface and risks falling into the sky. She meets a young boy who's gravity is regular. Together they must escape as they are hunted by their dystopian government.
- Cencoroll: An easy 30 minute movie about a boy and his mysterious partner that can transform. His young classmate accidentally finds out about his secret and gets abducted by a villain with a partner of his own. Watch this brilliantly animated short action movie!
- Sword of the Stranger: Classic Samurai movie with well--paced story and beautiful action sequences. BONES truly delivers in this classic movie about a samurai who calls himself Nanashi and the boy and his dog that run into him.
I may try to find some more recommendations later.