Here is the current list of anime you can request from~ Remember. This is only the current list until I watch more anime. And if you're wondering, this list doesn't include all the anime I've watched so far.
Angel Beats
Assassination Classroom
Attack on Titan
Baka and Test
Big Windup!
Black Bullet
Black Butler
Bleach
Blue Exorcist
Bungou Stray Dogs
Corpse Party
Corpse Princess (Shikabane Hime)
D. Gray Man
Daily Life of High School Boys
Darker than Black
Death Note
Death Parade
Fairy Tail
Fate/Zero Fate/Stay
Free!
Gangsta
Hakuouki Shinsengumi Kitan
Haikyuu!!
Hetalia
Hiiro no Kakera
Hotarubi no Mori e
Inu x Boku SS
Joker Game
K
Kamigami no Asobi
Kaze no Stigma
Kenichi: the Mightiest Disciple
Kuroko no Basuke
My Little Monster
Nabari no Ou
Nanbaka
Naruto
Noragami
One Punch Man
Ouran High School Host Club
Phantom: Requiem for the Phantom
PSYCHO-PASS
Reborn!
Samurai 7
Samurai Champloo
Sankarea
School Days
Sekirei
Sengoku Basara
Seraph of the End
Shiki
Soul Eater
Sword Art Online
The Future Diary (Mirai Nikki)
The Irregular at Magic High School
The Morose Mononokean
Tokyo Ghoul
Tokyo Ravens
Tsubasa RESERVoir CHRoNiCLE
Vampire Knight
xxxHolic
Yu Yu Hakusho
91 Days