Castle in the Sky (...ironically the very beginning because it was so pretty/depressing?)
Chi's Sweet Home (Chiiiiiiiiiiiiiiiiiiiiii!!! ;-; You'll see that black cat again! I hope!)
Cowboy Bebop (the videotape episode)
Fruits Basket
Fullmetal Alchemist
Ouran High School Host Club (end of the Wonderland episode)
Pokemon (because Butterfree leaving and Pikachu (almost) leaving was depressing to my elementary school age mind)
Princess Tutu
Romeo x Juliet (although to be honest I never had a chance to finish it...)
Sailor Moon (since I've been watching it since middle school there's probably more, but I think
The Girl Who Leapt Through Time