Approved Anime

This is a list of /tg/ approved anime, organized loosely into genres.

Add important details (e.g. tv series or OVA, number of episodes or movies) in brackets.

Science Fiction

Cowboy Bebop: Animu fans will argue forever if this is the best anime or not, but it is probably one of the best space-westerns anybody has made ever. An astounding look into one of the best parties an RPG could ever have. [tv series: season 1 - 26 episodes; 1 movie]
Ghost in the Shell: /k/ makes a show about cyborgs and politics. The two movies and TV series take place in different continuities, and have very different feels. You may love one but hate the other. [2 movies; tv series: season 1 - 26 episodes, season 2 - 26 episodes, 1 made-for-TV movie]
Serial Experiments Lain: The trippiest philosophy class you will ever take. A near-future set around 2020 or so, diverging around 1990. A must-watch for cyberpunk and conspiracy fans, as well as existentialists. Paranoia, Eclipse Phase, and Shadowrun all probably took a page from Lain's book.

Attack On Titan: Humanity is all but exterminated by the Titans, a race of tall creepy looking humanoids who eat humans for fun. Extremely graphic in the many and often undignified deaths throughout the series to almost 40k amounts levels of death per episode. Running themes are self-sacrifice for the greater good, the cheapness of life of the individual and Humanity Fuck Yeah. Main character is a short angry dude who wants to kill all Titans because they tore his mother in half and ate her. Is very well known for its kickass intro song.
Fist of the North Star: The manliest show ever made.
Tengen Toppa Gurren Lagaan: The Orkiest show ever made. [TV Series: 30 episodes; 2 movies]

Girls und Panzer: Just what it says on the title. WWII Tank warfare for sport, done by schoolgirls. You will make a Ōarai Girls' Academy themed Leman Russ tank company after watching it. The anime is now officially associated with World of Tanks.
Hikaru no Go: A young child is possessed by the Go obsessed spirit of a long dead Japanese gentleman. Epic gaming showdowns ensue.

Neon Genesis Evangelion: A philosophical character drama and Lovecraftian Horror Mindrape that pretends to be a mecha anime for its first half. Either one the greatest anime ever produced, or an overrated piece of tripe that collapsed under the weight of its own pretentiousness, depending on who you ask; there is no middle ground. Inspiration for Adeptus Evangelion. Obviously. [TV series: 26 episodes; 5 movies]

Berserk: Warhammer Fantasy Roleplay: The Anime. Guts, a brutal and unstoppable swordsman, walks the land of grimdark as he recounts his impossibly bad-assed past. Noted for being GUTS HUEG because HUEG GUTS has HUEG GUTS.
Record of Loads of War: ADnD: The Anime.
Slayers: 2E: The Anime.
Spice and Wolf: A show about Horo, wolf-girl pagan goddess of the harvest (Often mistaken for Leman Russ.), and also economics. Proof that not all medieval fantasy has to be sword-and-sorcery to be interesting. [TV series: Season 1 - 13 episodes, Season 2- 13 episodes]
Strike Witches: (Little Girls in Panties): WWII flying aces redrawn as loli airplane machines which zap aliens while flying around without pants. Not really beloved by /tg/, but someone thought something about it would make a good homebrew.

Haiyore! Nyaruko-san: It's one of those wacky highschool comedy bits that Japan shits out every season, except starring Nyarlathotep. Yes seriously. Pop culture references, /d/eviance, Sanity loss (complete with Call of Cthulhu character sheets), and gratuitous canon rape ensue. "She's an Eldritch Abomination, not your waifu!"
Fireball: And the second season, Fireball: Charming. A CGI piece about robots made by Disney Japa-Wait, no, don't go!! It's full of absurdist humor, and every kind of off-kilter pun, all while being a precisely accurate depiction of the Men of Iron incident from Warhammer 40,000.

Homestuck: Despite being a webcomic, people often claim Homestuck is their favorite anime. Strangely, the Homestuck fandom has countered with an attempt at creating an actual Homestuck Anime; the fandom is known for their ability to get shit done, and quickly, but only time will tell if the project goes anywhere.
Touhou, which is not actually an anime or manga but is frequently confused as such by those unfamiliar with it. It has however, received several official manga spinoffs.
Wakfu: A French cartoon about a kid named Yugo who discovers he is part of a long-lost race of people with the ability to create portals. A fun world with fun characters and a surprisingly deep Big Bad. Not to be confused with your waifu.
Avatar: The Last Airbender