The Villain Must Be Punished

Under most circumstances, the heroes are perfectly content with thwarting a villain's Evil Plan. This is not one of those circumstances; the villain has harmed innocent people, performed all manner of horrible deeds, kicked every dog out there. Simply stopping his plans isn't enough anymore; whether by humiliation, death, or worse, the Big Bad has to actually pay for what he's done.

Compare It's Personal, a common reason why the heroes come to feel this way; and Laser-Guided Karma, where the universe itself does the punishing. Contrast with Karma Houdini. Can sometimes trigger (or just be another point in) a Cycle of Revenge.

Anime and Manga
"Ash: I'm not done with you yet..."
 * Dragon Ball Z:
 * During the Frieza Saga, even as Namek is about to explode, Goku makes it his mission to finish the fight with Frieza against his friends' objections. It isn't enough that he's thwarted Frieza's attempt to use the Dragon Balls for gain immortality; Frieza has to pay for everything he's done, and Goku has to ensure he does so, beating him to a pulp and ripping his pride to shreds before deeming him Not Worth Killing.
 * Brutally Deconstructed during the Cell Saga. After reaching Super Saiyan 2, a rip-roaringly pissed Gohan beats Cell to a pulp, but when given the chance, opts to stand back and let Cell regenerate just so he can continue the beatdown, feeling Cell deserved to suffer as much as possible before he's done. This backfires when Cell, driven to a Villainous Breakdown, attempts to self-destruct and take the Earth with him, forcing Goku to sacrifice himself to stop it.
 * Pokémon: Ash and his friends generally defeat the Team Rocket trio rather easily and let them flee, but there are times where their antics push them past their Rage Breaking Point, after which Ash makes sure that they're not leaving unless he sends them blasting off again.

Comic Books

 * The Killing Joke: Joker invests a lot of time in tormenting Commissioner Gordon, intending to break him and prove that "one bad day" would drive anyone crazy just as it did him. Batman saves him, after which Gordon insists that Batman go after Joker and bring him in.
 * Punisher's entire schtick revolves around this. He doesn't want to just lock crooks up; he wants to make them pay for their crimes with their lives.

Live-Action TV

 * Buffy the Vampire Slayer: Warren Mears, the Big Bad of Season 6, quickly shows himself to be a vile excuse for a human being, using a Mind Control Device to turn his ex-girlfriend into a Sex Slave, killing her when she snaps out of it and tries to escape, and using magic and time-distorting demons to dupe Buffy into thinking she killed Katrina so she'll take the fall, which nearly succeeds. By the final episodes of the season, he's shot Buffy, nearly killing her, and killed Tara, the latter of which unleashes Dark Willow, who fully intends to kill Warren for it. Dawn and Xander are so disgusted and furious with him that they openly support Willow's intentions; Buffy, meanwhile, preaches Thou Shalt Not Kill, but only because she doesn't want Willow to become a murderer.

Video Games

 * Banjo-Kazooie: After saving Tooty by completing Grunty's sadistic Game Show, Banjo and Kazooie go back home and have a cookout with Bottles and Mumbo Jumbo to celebrate their victory... until Tooty reminds them Grunty escaped and demands they go back and take her out. Only once Grunty is knocked off her tower and trapped under a boulder is Banjo allowed to kick back and relax.