I always have trouble choosing a favourite film, I can never really decide. Likewise, I really can't decide which of these anime movies and series are my favourites. They are all amazing. I watched most of these with my boyfriend, who introduced me to anime.
Neon Genesis Evangelion, The End of Evangelion and Evangelion OVA
Hellsing and Hellsing Ultimate
Ghost in the Shell 1 and 2 (movies, I need to see the SAC)
Howl's Moving Castle
Code Geass: Lelouch of the Rebellion
Bubblegum Crisis Tokoyo 2040 - plus that one of it's sequels I can't remember the name of
Memories (Shorts: Magnetic Rose, Stink Bomb, Cannon Fodder. Magnetic Rose is particularly beautiful.)
Cowboy Bebop movie (seen a few of the series, pretty good too)
Final Fantasy: The Spirits Within (although it's not exactly anime)
Goku: Midnight Eye (only 2 episodes ever made. The first was particularly amazing)
Vampire Hunter D
Vampire Hunter D: Bloodlust
Fullmetal Alchemist Brotherhood? Not sure if this was the one I watched, it was years ago, and in the middle of the night, but I remember thinking it was great. Can't remember the plot.
The Castle of Cagliostro (Lupin III) I love the French movie about the original Lupin, too!
Parasite Dolls?: Can't remember if I've seen it or not.