Majin Buu of Dragon Ball Z had the mentality of a child without a conception of right or wrong, so he quite cheerfully obliterated anything that wasn't entertaining. After Mr. Satan befriended him, Buu slowly began to evolve a sense of morality.
Taken even further when Buu's pure form is revealed: Kid Buu. This form is hinted to only have as much power as Super Buu, but lacks any of the restraint displayed by his other incarnations. He's a complete psychopath, as demonstrated when he destroyed the planet he was standing on.
Peach Girl: Momo is bullied by Sae throughout the course of the story, because Sae is jealous of her for getting more attention and hates her, though Sae often lies to people about this, trying to convince people that it is *Momo* who is bullying her. And she succeeds for a while before she finally gets what's coming to her. Also Momo is put down by her classmates and other people for having tan skin and bleached blonde hair, because they believe that she is a major party girl who spends all her time tanning at the beach; they also sometimes insulted her by calling her a slut because Sae tricked them into thinking that Momo stole Toji from her when it was really Sae who was trying to steal Toji from Momo, which, in reality, is entirely wrong, as Momo is actually a really nice girl, it's Sae who's the Manipulative Bitch.
Candy's step-siblings Eliza and Neil Reagan in Candy Candy. They are so utterly cruel, that it's unthinkable to even feel a twinge of admiration or sympathy towards them. Worst is the Rich Bitch Eliza, whose motivation for making Candy's life hell is: because she can. If that doesn't convince you enough, wait until you see the episode about Eliza getting temporarily grounded, whereas she will make up a monologue that explains that she really loves violence, especially towards people that can be picked on, like Candy. So she not only did it because she can, she also LOVES and takes joy in doing it.
Tetsuo from AKIRA. In a scene after Tetsuo, in the full throes of his psychic megalomania, kills fellow gang member Yamagata, Kaneda reflects on how everyone in his school had teased Tetsuo and tried to make him cry.
Narutaru contains Aki Honda and her Girl Posse, who manage the rather impressive feat of turning schoolyard bullying into a monstrous act when they rape one of the side characters, Hiroko, with a test tube; and as if that's not nasty enough, the original plan was to kick Hiroko in the stomach while the test tube was inside her!. Once Hiroko obtains her dragonet, she gets her revenge... in very bloody fashion.
Gaara of Naruto. Yes, make the kid who contains a gigantic malevolent sand demon upset. That's definitely going to score points.
This could've happened to Naruto as well (since Gaara is Naruto's Evil Counterpart), but he never actually did anything about it because he never knew why the adults were treating him like the plague. Thankfully, none of his classmates knew either.
Death Note has Teru Mikami, whose childhood involved having to deal with asshole kids. As a result, when Mikami gets his hands on the titular Artifact of Doom in the second half of the series, he develops the philosophy that past middle school, anyone who commits a cruel act is basically unredeemable and should be, to use his favorite term, "deleted." By the end of the anime, he's become the most psychotic of all the Kiras, with the much-contested title of "Craziest Character in the Series."
Bokurano has several examples of this trope as well — the major victim would be Mako Nakarai, who is bullied mercilessly because her mother is a prostitute.
Kodama is a protagonist version. He cares little for other life (he says that if something dies it's their destiny, shoots cats regularly, and in the anime boils a crab alive using a firecracker), to the point where he kills dozens of people while driving the mecha. He accidentally kills his father, sending him into a Heroic BSOD, right before he dies. Ushiro also counts, being easy to anger and quite violent toward his younger sister.
Flame of Recca is all over this trope. For most (if not all) of the characters with tragic pasts (such as Fuuko, Kaoru, Kurei, and many others), a good part of said past was being tormented by other kids, who take it to extreme heights for seemingly no reason other than the fun of being evil little bastards. One of the tormented kids (Kaoru) was even driven to attempt suicide, and a few others considered it.
Kaze no Stigma: Kazuma was born into a prestigious family of fire users yet had no talent in the arts at all, not even the standard immunity to fire that even the weakest branch members had. Consequently, as a child, he was regularly beaten and tortured with fire by the other children, ranging from 5 year olds to teenagers. At least one time, they were seriously going to kill him.
Mad Pierrot from Cowboy Bebop has regressed to having a child's mind from sadistic experiments, and as Jet puts it, there's nothing as purely cruel as a child.
Toyed with in Ranma 1/2; when Ryoga Hibiki and Ranma Saotome were kids, they went to the same all-boys Junior High, which had a policy of simply flinging out the stuff from the lunch shop and letting the students fight over it. Ranma would consistently beat Ryoga when they went after the same piece of food, but otherwise got on with him okay enough that he would regularly lead Ryoga to and from his home, to the point Ranma still remembers where Ryoga lives better then Ryoga does. Eventually, Ryoga got sick of Ranma beating him in combat and challenged him to a duel; when he took four days to get there and found Ranma had left, he set off after him to try and get that match. Initially, however, he didn't seem to really be mad at Ranma, more wanting to settle the score. When he wandered into Jusenkyo, however, he fell into the Spring of Drowned Piglet in an incident involving a girl and a panda, and he promptly blamed Ranma for it, becoming much more furious and even declaring he wanted to kill Ranma. Things kind of peaked when he found out Ranma was actually at fault, but it quickly petered out; much as they squabble and bicker, the two watch each other's backs and save each other's lives on a regular basis.
The Drifting Classroom is a prime example of this. When the school is transported into a Crapsack World and the kids get more and more desperate they start doing horrific stuff such as killing each other with spears and knives for food, burning a fellow student on a cross, and even (EATING A DEAD STUDENT'S REMAINS.)
Umineko no Naku Koro ni: This is Rosa's justification for her treatment of Maria (or what she tells people anyway; in Maria's backstory we find out that Rosa hates her a good deal for just existing). Her excuse is basically, "All the children make fun of her! Don't you see?! Beating her will obviously make her stop whining!"
Bern and Lambda as well.
Not to mention Ange's classmates.
Due to the nature of the series though, some of these examples may be subverted. An important theme in the series is differences in point of view, so what a character perceives may not actually be true and can't be trusted. For example, there are a few hints in-series that suggests Ange's bullying was all in her head or over exaggerated.
Episode 10 of Excel♥Saga had Menchi get chased by a mob of kids all trying to pelt her with rocks, apparently for no reason other than the fact that she was there. Not to mention that the only one that talks is a little potty-mouth.
Rosario + Vampire: Moka Akashiya was teased and bullied because of her vampire origins when she was a kid.
Deadman Wonderland: It must be something that in a world of super-powered sadists some the cruelest characters are muggle bullies whose victims later became the twisted un-powered (or at least not till recently) playthings/enforcers of the prison. Examples include kitten killing, possible rape, and playing Othello on a kid's back with thumbtacks. And this is before the devastating earthquake that plunged the land into chaos.
In terms of just plain Ax-Crazy, Little Hibana killed a classmate while in kindergarten; Minatsuki killed her mom for abandoning her during the earthquake and her dad just to see if her brother was gullible enough to think their loving dad would try to rape her (he's not, he just has a huge Big Brother Instinct, which she finds incredibly comforting). Minatsuki's current pastimes are sadism, Blood Sports, and gardening.
The children of Ohara were like this to Nico Robin in her tragic backstory in One Piece.
Wrath from the first Fullmetal Alchemist anime is an extreme embodiment of this; his childish, emotionally underdeveloped nature makes him a vicious tool of destruction, as seen when he sadistically kills Lust.
Marie's classmates in Higanbana No Saku Yoru Ni call her a youkai to her face, boys flip her skirt up, and she's usually forced to do all the cleaning herself. One scene has a girl stab Marie's hand with a pencil for mumbling "Oh, really?" during a group discussion, saying "Youkai shouldn't talk."
The kids of SoilNew Town aren't as perfect as they appear. "She was our friend" is actually "we talked occasionally but she was a weird outsider and then she fell into a narcoleptic sleep so we left her in the middle of the road to be almost raped" while another group of boys ID their mysterious leader from the burn where they set his hair on fire. Then again the boys have an excuse, considering they're all current victims of their pedophile rapist dentist (their real leader, a high schooler, is also a victim, as are the slightly older rapists who raped his friend), although they don't know the connection, they just want to destroy this lame world.
Hanseland Gretel are probably the most psychopathic examples of this you're likely to find. They kill everyone in a bar (even though they were just targeting two people), tortured a man to death, and being insane sociopaths in general. They probably weren't always like this though.
In Sangatsu No Lion, Rei heavily implies that he was bullied in his younger years both in a passing statement about his school life in comparison to his home life and in his insistence to Hayashida-sensei to not spread the fact that he's a professional shogi player to his new classmates.
In Code Geass, Lelouch was often the victim of this from young Japanese kids in his earliest days there, including his would be best friend, Suzaku.
Most of the Members of the Zodiac with Implausible Hair Color got teased or bullied for it at some point. Most of them found ways to retaliate and stick up for themselves, but poor little Kisa was so traumatized, she stopped talking for a long time.
Koe No Katachi is all about the cruelty of children and the main focus is on a character who has been driven to suicidal edges by a mix of the treatment he receives from his fellow students and the memories of the way he constantly attacked and abused a deaf girl in his class when he was younger. The story focuses on his attempts to redeem himself with the girl and somehow makeup for his actions.
Another disability-related example, With The Light gives a lovely example of kids mocking Kanon because she has a "stupid" autistic brother, not stopping even when lectured until Kanon fought back.