TV Series / OVAs
Romance and Perversion: Chobits, Love Hina
Romance and Action: Inu-Yasha, Tenchi Muyo! (OVA only)
Action: Trigun, Cowboy Bebop, Noir
Feudal Japan and Action: Kenshin Imura
Complete Craziness: FLCL, Excel Saga (Parody)
Craziness and Action: Nadesico, Jubei-Chan, Slayers
Giant Fighting Robots Sci-Fi: Gundam Wing
Giant Fighting Robots Fantasy: Escaflowne
Fantasy: Record of Lodoss War
Sci-Fi "Ouch my brain" Weirdness: Serial Experiment Lain, Neon Genesis Evangelion
Movies
Akira
Mononoke-hime (Princess Mononoke)
Naussica (don't get "Wind Warriors")
Spirited Away
Vampire Hunter D
Tenchi Muyo in Love
Tenchi Muyo 2: Daughter of Darkness