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Tear Jerker: Blue Exorcist
Blue Exorcist starts out with the protagonist's adoptive father, Fujimoto, committing suicide to prevent Satan from taking him away. Later on, he and his brother find Fujimoto's familiar, a cait sith named Kuro, who has gone berserk upon hearing about his death. He refuses to accept it until the protagonist head butts him and offers friendship. Kuro breaks down into tears as he realizes his friend is truly dead.
Fujimoto's death got taken Up to Eleven in the anime adaptation.
When Kuro is in furious denial of Shirou's death and has become a danger, Rin manages to convince him of the truth. Kuro's grief-stricken acceptance of that truth is heartbreaking, to say the least.
Also in the anime's special, Kuro leaves home in search of Shiro's grave, and imagines his spirit talking with him and consoling him. It's depressing, to say the least.
Usamaro the rabbit demon in The Movie. All he wanted was for everyone to be happy and forget their sad memories, and he ended up being sealed away for it.
And at the end, his Heroic Sacrifice and the fact that Rin is the only one who remembers him.