Some of the greatest moments in anime come when characters are pushed to their limits. Whatever is happening to them is so upsetting that their usual coping mechanisms or battle tactics just aren't cutting it.
Among the best anime characters who snapped who are those who unleashed an attack that no one thought they were capable of, like Gohan when he decides to stop sitting idly by and beat the tar out of Frieza. Others don't get physically violent, but verbally eviscerate another person in a way that's completely unlike them, like Nadeko Sengoku does in Monogatari.
Which characters have the most awesome moments of losing it?
If there's one thing that Envy taught us in Fullmetal Alchemist: Brotherhood, it's that you shouldn't piss off Roy Mustang. After walking in on the homunculus strangling his "irreplaceable subordinate" Riza Hawkeye, Mustang loses it and pummels Envy over and over with deadly explosions. His final attack is so powerful it causes Envy to slump over and reveal his powerless true form - a small caterpillar-like creature that is helpless without a host or Philosopher's Stone to draw power from.
If anything is likely to make a person snap and go berserk, it's being tortured. After enduring everything from having his fingers and toes chopped off and regenerated over and over again to having a centipede dropped into his ear, Ken Kaneki just can't take it anymore.
The kindhearted young man he once was disappears, and his hair turns white. Where before he preferred to avoid violence if at all possible, now he's fully willing to fight for his life. But he goes much further than just fighting - he takes a bite from Yamori's flesh and eats his kagune, leaving him a horrifically injured wreck before escaping.
Naruto's nine-tailed fox form is unleashed whenever he sustains some form of extreme emotional pain. This happens several times throughout the series, especially toward the beginning when he's not on good terms with the tailed beast inside of him. Tailed beast mode makes him extraordinarily powerful, but it also means that he isn't in control of his behavior and is more likely to hurt people in ways he otherwise wouldn't.
Activating tailed beast mode is dangerous because it requires his cells to rapidly replace themselves, thus shortening his lifespan. Even worse, when the mode is active, the tailed beast could escape and wreck havoc. The form is activated several times, but one of the most dramatic times occurs when Hinata seems to have perished. When this happens, Kurama nearly escapes from his seal.
Gohan is a fairly mild-mannered child in Dragon Ball Z, but seeing his friends and family being hurt by a villain often leads him to snap.
After Frieza gores Krillin with his horns and tosses him into the ocean to die on Planet Namek, Gohan desperately wants to save him. Frieza blocks Gohan's path and taunts him, causing Gohan to go berserk and start pummeling the Saiyan-hater with a power level that shocks everyone, especially Vegeta. Unfortunately, Gohan's attack wasn't enough to completely beat Frieza, but it certainly was a preview of things to come later in the series.