In the film version of Nausicań of the Valley of the Wind, Princess Kushana has lost an arm (and possibly more body parts) to the insects of the Sea of Corruption. Hence her attempt to wipe out the Ohmu with the God Warrior. This subplot — origin, motivation, and outcome — doesn't exist in the manga.
Grandpa Gohei and Akakabuto, the bear who took his left ear and killed his hunting dogs in Ginga Nagareboshi Gin.
Quent Yaiden from Wolf's Rain obsessively hunts wolves, believing they are responsible for the death of his family. Actually, it was the werewolf Lord Darcia who burned Quent's village to the ground to flush out true wolves, and ironically, it's the wolf Toboe who later tries (and fails) to save Quent from Darcia.
This trope is pretty much the premise behind Teenage Mutant Ninja Turtles in all its incarnations, with the human Shredder being the mortal enemy of the eponymous reptilian heroes. In almost every version, this also applies to Splinter himself, who was originally Hamato Yoshi's pet rat. After seeing his owner murdered by the Shredder, Splinter became determined to avenge him. The 1980s cartoon averted this by merging Splinter and Yoshi into one character, essentially having the human Yoshi be mutated into an anthropomorphic rat instead of an animal that gained human traits.
The original The Transformers comic did a Whole Plot Reference to Moby-Dick at one point. The Transformer in question lived on a world far from most of his race and was a Pretender (a Transformer covered in a pseudo-organic shell that allowed him to pose as an organic but was rather difficult to repair) and so couldn't get his leg properly restored. He finally incapacitated the beast responsible but decided against finishing it off.
Kevin the Flightless Bird from Up. Justified in the animal was part of an undiscovered species and the antagonist, who had been discredited by the scientific community, wanted the recognition for finding it.
Inverted in The Jungle Book; Big Bad Shere Khan (a tiger) actually wants to kill Mowgli just for being a human.
Amos Slade (and, to a much lesser extent, Copper the hound dog) during the second half of The Fox and the Hound, who wants to kill Todd the fox out of revenge for (indirectly) injuring his dog Chief by having him be hit by a train. Originally, he was actually going to be more so, as Chief was originally going to die after being hit by the train.
In Of Unknown Origin, a New Yorker becomes so obsessed with eliminating the pesky rat infesting his apartment that he eventually trashes his own place in frantic pursuit of it.
Invoked in Soul Surfer when the shark is shown harpooned and dangling as a trophy. Apparently, Bethany Hamilton's surfing buddies think that when you mess with one of their own, It's Personal.
In Immortal, Inspector Froebe has had a grudge against the Dayak ever since it bit half of his face off.
The B-movie Orca: The Killer Whale, which is about a vengeful, bloodthisty orca who wants to get revenge on humanity after its mate and calf were both killed by fishermen. Before it begins its rampage at a nearby coastal city, the first thing it kills is a great white shark...
In Dragonheart, the knight Bowen becomes a dragon slayer after he (incorrectly) comes to believe that the dragon Draco is responsible for corrupting the heart of Einon. He eventually hunts dragons to near-extinction in his search for Draco over the years, and even after realizing his the error of his ways has no choice but finish the job to stop Einon's evil for good.
Ahab was referenced in the movie Star Trek: First Contact, when Captain Picard brutally guns down several Borg in the holodeck and refuses demands to self-destruct the Enterprise in lieu of stopping the mechanical menaces personally, all because of the violations he suffered when they assimilated him and his anger over Starfleet's inability to halt their encroaching on Federation space. Lampshaded when the 21st-century woman referencing the book admitted that she hadn't actually read it and only remembered the idiom that went with it. This was acknowledged by studio execs, soon after the release of that movie, Patrick Stewart played Ahab in a made-for-TV movie of Moby Dick, mainly because he was so good at a referenced state of mind that they decided he would be good for the real thing.
Older Than Radio: One of the best known examples is the deadly rivalry between Captain Ahab and Moby-Dick. The relationship has been subject to much parody. Ahab himself hung a lampshade on this trope in the original novel — he essentially said that if he couldn't hunt down the abstract fates that ultimately cost him his leg, the eponymous whale would be the next best thing.
Peter Pan has the crocodile who gave Captain Hook his nickname. This might be a bit of an inversion, as it is actually the crocodile who is violently obsessed. In Hook, we learn that Captain Hook finally did hunt down and kill the crocodile, stuffed it, and put it on display as a giant silent clock. That'll teach it. (Of course, he tempts fate by fighting underneath it. The (in)evitable happens, the clock tower topples over and the crocodile's jaws falls on Hook, effectively ''eating him''.)
As indicated in the page quote, this was Prince Rilian's undoing in The Silver Chair. Yeah, the snake that killed his mother was really a Scaled Up mind-controlling witch and thus perfectly capable of forming intent, but he had no way of knowing that.
Weird example in Warhammer 40,000 novel Storm of Iron: an Iron Warrior captain imagines that he is hunting a legendary beast while trying to kill a Warhound Titan (Warhounds are the smallest of the Titans, with a focus on infantry/armour support role rather than a "level cities" role).
The Western novel Killers of Man by Ralph W. Cotton features a wealthy businessman determined to get revenge on the grizzly bear that took his leg.
Parodied in Railsea by China MiÚville, in which Captain Naphi is perpetually hunting Mocker-Jack, the blonde giant mole that took her arm. In the community Naphi comes from, every real hunter is required to have an Animal Nemesis, and it turns out that she didn't actually lose the arm.
In the Grand Finale, he finally catches it after tearing down half the school. Then it turns out to have had babies, so they adopt it as a mascot.
Sawyer in LOST has managed to do this on two separate occasions. Once chasing after a boar who ransacked his tent and then chasing after a treefrog which annoyed him with its croaking. He eventually caught up with both of them with help from Kate and Hurley respectively. He decided to let the boar go as it was "just an animal" but then rejected Hurley's offer to relocate the frog and simply crushed it in his hand.
Happens in an episode of Desperate Housewives in which Lynette spends half an episode hunting down a possum because Possums = Cancer.
Al Bundy went through this in one episode of Married... with Children, becoming obsessed with a rabbit that ruined his vegetable garden and trying to kill it. Of course, Hilarity Ensues, especially given that the rabbit has the intellectual advantage.
An episode of Breaking Bad has Walter White acting bizarrely, obsessed with killing a fly that's somehow entered his laboratory.
Paul Phoenix and Kuma from Tekken have a version of this; neither actually says that they're rivals but the back story makes it pretty clear.
Final Fantasy Tactics A2 has a sidequest revolving around a clan named House Bowen, whose leader seeks revenge on a gigantic cockatrice for killing his wife.
The game Cabelas Dangerous Hunts 2009 is about a famed big game hunter who swears revenge on a bear who killed his best friend.
Skies of Arcadia features an old fisherman named Drachma who chases the giant flying whale Rhaknam for killing his son decades ago. Like the rest of the game, it's an old tale/genre/mentality (obviously Moby-Dick in this case) Recycled InAir.
In The Sims Medieval, your hero can be given the "Whale Ate My Parents" trait, which, er, speaks for itself. The result is that they will become obsessed with whales to the point of occasionally getting a "Whale Rage" buff.
Peasant's Quest, which is apparantly about a young peasant in short pants named Rather Dashing who is obsessed with killing the evil dragon Trogdor the Burninator out of revenge for burning down his thatched-roof cottage, and in order for him to do so, he must dress, smell, and be on fire like a real peasant. The game ends with Rather Dashing finally confronting Trogdor only to find out that Trogdor can never be killed no matter what, and as a result, Trogdor decides to kill him anyway.
In RuneScape, players can meet a whaler named Hubbub, whose ship was destroyed and crew killed as he tried to kill the blind white whale named Shuma, in revenge for her destruction of human villages in the Eastern Lands, including his own.
Fantasy example from 8-Bit Theater: Garland and his almost incapacitating hatred for Forest Imps. Admittedly, that is what they want you to think.....
It's usually Bun-bun from Sluggy Freelance that wants the revenge on humans, but he and Santa Clause have a mutual feud.
Belkar of The Order of the Stick isn't exactly human himself, but he has a similar grudge against Windstriker, Miko's horse. Said horse was one of the characters he asked the Oracle if he'd wind up killing, and he was quite disappointed when he discovered that Miko had died, leaving Windstriker in the Outer Planes, meaning he'd never be able to kill Windstriker.
The Adventures of Dr. McNinja parodies this with the birdosaurus, a dinosaur that follows the doctor around and performs minor actions with more serious consequences (e.g. stepping on the stone that sets off all the traps while the doctor is standing in the trapped area), which he has developed an intense, somewhat disproportionate hatred for.
Doctor McNinja: I hate you. I hate you so much. I will hunt you down forever. When you die, my laughter will be so bright, it will be the last thing you will hear, see, smell and feel.
Oglaf: This strip has an entire ship of obsessive hunters who are hunting the great white whale which took their legs.
Elmer Fudd and Bugs Bunny in Looney Tunes. And Yosemite Sam and Bugs Bunny. And Rocky the Gangster and Bugs Bunny. And...well, every human character to ever interact with Bugs Bunny in more than one short.
There was a character in Disney's Aladdin: The Series that was obsessed with hunting down some sort of desert whale/shark, in what was obviously a homage to Moby-Dick. Though his quest was actually quite practical, as the belly of the sand shark was covered in precious jewels.
But when he does catch it, he has no idea what he will do now, so he decides to cut the shark free and hunt it again.
The Simpsons has an episode where Homer is menaced by a bear and constructs a metal "safety suit" in order to kill the creature and regain his respect, which elicits the page quote. However, this gets subverted when Homer realizes the bear is lashing out in pain due to an electronic tracking tag on its ear, so he removes the tag, ends up befriending the now gentle bear, and gives it the suit so it can brave a gauntlet of hunters to get to a wildlife preserve. Which sparks a feud between the bear and a particularly mean-spirited elephant.
In Family Guy, Peter Griffin's Arch-Enemy is a giant yellow chicken that was introduced on the show in a very elaborate homage to many action movies.
Much like the literature example above, the Crocodile that ate Captain Hook's hand is still around in Disney's Peter Pan. This time around, though, while the Croc will gladly take a bite out of Hook if the opportunity presents itself, the Croc isn't totally fussed about getting at Hook and is happy simply screwing with him.
One episode of Dragon Tales had the main characters help a sky pirate hunt down a giant flying whale for eating his ship. At the end of the episode, they successfully capture the whale, and it turns out that the ship said whale ate was a toy ship.
Perry the Platypus from Phineas and Ferb is the Animal Nemesis of the resident Mad Scientist Dr. Heinz Doofenshmirtz. In one episode, after failing to convince Perry to show up and thwart him, Doofenshmirtz instead has a battle with a potted plant. It ends in a draw.
In the opening of Dan Vs. Anger Management, Dan is about to launch a nuclear warhead that will cause World War III at a family of squirrels he's currently upset at.
One episode had Generator Rex vs. an E.V.O. bunny. It kept going outside necessity to attack him.
One episode of The New Adventures Of Huckleberry Finn had Huck, Tom Sawyer and Becky Thatcher getting shanghaied by a crazed Ahab-like whaler who forces them to become his new crew in his forty-year mission to find and kill a monster whale, an obsession which eventually causes his doom and almost our heroes as well.
Predators sometimes, for whatever reason, will imprint upon humans as prey and go out of their way to hunt people even over what they would normally hunt. The only way to deal with such animals, known appropriately as Man-Eaters, is to invoke this trope, hunt them down, and euthanize them.
The most notorious and well known example are the Tsavo Maneaters. They were killed by Lt. Col. John Henry Patterson, who went full Captain Ahab to bring them down.