Certain players in most multiplayer PvP games are like this, especially when time-to-kill is low. Get their attention and they will chase you constantly until you are within range and try to kill you. They'll even prioritize this over any game objective.
Alma from the Adventures of Lolo games. She'll normally mind her own business, but if Lolo should get within her field of vision, she'll roll after him until either she catches him or he takes refuge in a meadow.
The good old xenomorph in Alien: Isolation, which will determinately chase Amanda Ripley across an entire gigantic space station, constantly showing up to plague her even if she's taken in-game traincar-like transport miles away. Or, considering it is ultimately revealed there's a whole nest of aliens infesting said space station, she might just be encountering a whole bunch of them.
While on the Frontier in Assassin's Creed III, you encounter wolves and bears which attack you on sight. Contrary to what happens in real life, the wolves are super persistent and attack you... even after you kill their alphas. They don't retreat even when it's in their best interest. Possibly justified since it's a computer simulation in-universe rather than "real" wolves.
Certain enemies in The Binding of Isaac such as Gapers and Globins do nothing but charge relentlessly at the player (and even after beating the latter, the player still has to destroy their remains to keep it from rising up again and resuming their charge.) Lust and Super Lust are the boss version of this, with more health and a much faster speed, to the point that characters with too low a speed stat just can't outpace them.
The giant... black... troll... thing, from Castle Crashers. It just won't stop chasing you until it's dead, no matter how many arrows (or magic attacks) you throw at it.
In Contagion (Monochrome), the Riot zombie is a zombie clad in bulletproof armor. It soaks up bullets like a sponge does water, and it is ridiculously persistent, chasing survivors throughout the entire map until it (the pursuer) is killed. They can be extremely difficult to deal with in harder difficulties where ammo is limited and zombies can kill in a minimal number of hits. The only easy way to get rid of it is to blow it up with an explosive weapon.
The walrus chef in Crash Twinsanity, who chases you halfway across N Gin's battleship — over seemingly arbitrary holes in the floor, through walls of crates — even nitro crates don't slow it down.
In Darkest Dungeon, if you encounter the Shambler and run from it, it will turn up in every future encounter until you kill it, you leave the dungeon or it murders you. (If you had to run from the Shambler once, odds are strongly in favour of "it murders you", so that abandon button might start to look very tempting.) Justified; the Shambler is an Eldritch Abomination that dwells in a weird cosmic dimension and isn't bound by the limitations of the physical world, and so it's honestly fairly generous of it to stop pursuing you at all and not follow the adventurers into the Hamlet and mug them in their rooms at the Sanitarium or the brothel.
Any regenerating Necromorph in Dead Space. They will hunt you, they are invincible and thus walking bullet sponges, their slow but unstoppable approach makes the player nervous, and they tend to be encountered in rooms with timed lockdown. Fortunately, you can get rid of them, just not so easily. Hunter, Übermorph, and Regenerators are stopped by shuttle engine, outrunning, and blasting to pieces, respectively.
The T-Rex in Dino Crisis will never stop stalking you. Every time you encounter and defeat it, it'll just back off for a little while until it's ready to attack you again. Naturally, you put it down for good in the finale.
Similarly, Dino Strike Wii has a T-Rex as it's last boss, who relentlessly comes after you after you empty what seems like a dozen grenades down it's throat. It takes an erupting Chekhov's Volcano finally drowning it in lava to finally kill it.
Dynasty Warriors 8 has the supremely powerful Lu Bu who, except in specific missions where the entire point of the mission is to defeat him, is always immensely buffed to the point of effortlessly crushing everything in his way. While he does come after your forces in certain missions, if Diaochan is on the same map and the player defeats her, Lu Bu will become so enraged that he'll enter his buffed state if he isn't already and specifically target the player for the rest of the mission, which is frequently followed by a swift and horrific death at his hands unless you can play keep-away long enough to complete the mission.
Morrowind and its Bloodmoon expansion play this straight. Both Vvardenfell and Solstheim feature More Predators Than Prey, and it's quite difficult to lose an enemy creature if aggro'd. However, the notorious Cliff Racers are even more difficult to lose thanks to their ability to fly.
Every enemy is this. That mountain lion, that wolf, that troll, that rat... will chase you down to the ends of the earth in order to maul/feast upon your flesh. They will come after you through villages, forests, rivers, lakes, mountains, and plains, from the farthest western point of the map to to the farthest eastern point, in an all-consuming, single-minded drive to wreak vengeance on you for entering their line of sight.
The King of Miscarcand (a powerful undead boss fought in the main questline) has the trademark persistence of an Oblivion enemy, but (due to a bug) he never loses track of the player's position, and he's not afraid of entering crowded cities.
Skyrim keeps this tradition alive in the series. Predatory creatures including bears, wolves, and sabrecats will chase you halfway across Skyrim once aggro'd.
Etrian Odyssey: Once you've triggered an F.O.E. encounter, you won't get rid of it until you kill it or it wipes your party. Some F.O.E.'s are also programmed to start coming after your party as soon as you step foot into a dungeon, even if you haven't encountered it.
The Monsters of Evolve are this in spades. Massive extraterrestrial predators at least as smart as humans, they have an almost pathological drive to hunt and kill people. Humans too well equipped to kill? They'll hide in the wilderness, feed on other animals, grow, and repeat until that's no longer true.
The Fallout 4 mission "The Devil's Due" introduces possibly the most persistent Deathclaw in history. Who tracks a group of Gunners across a huge swath of territory to retrieve its stolen egg.
Being a fairly early MMO, mobs in Final Fantasy XI originally had no tethers. Once aggroed they would chase you indefinitely until you crossed a zone line, and if they belong to a species that could link they would bolster their numbers with any others they passed. Once their prey crossed the zone line they would then slowly wander back towards their spawn location, aggroing anyone else who had the misfortune of crossing their path. Popular EXP zones like the Jungles, Garlaige Citadel, and Crawler's Nest were easily paralyzed if one group pulled something high level they shouldn't or linked far too many mobs at once. Every group in the area would have to flee the zone and wait for an "all clear" from a dead person still inside once everything had left the area. Anyone zoning back in too soon would just pull the danger back to the zone line. Mercifully, Square finally tethered some mobs to despawn if pulled too far, and any mob that loses its aggro at a zone line despawns to eliminate the danger to bystanders.
Many of the monsters in Final Fantasy XII seem to go to rather crazy lengths to catch the party. The ones that teleport, however, won't leave you alone until you leave the room, and a few monsters might just keep going.
Most enemies in Final Fantasy XIV will give up chasing you after you run away for a while. This makes fair sense for things like animals, that were likely just guarding their territory, and are placated by leaving them alone. It makes less sense when you run right past a guard to an enemy military base, and he stops chasing you once you're 50 yards inside the base. Guess it's not their problem anymore.
Played straight in dungeons, where enemies that spot you will hunt you down no matter how much you try to escape. This is actually beneficial in most instances, as now players can have a tank pull a huge group of mobs from a broad swath of the dungeon, group them up, and then take full advantage of AoE attacks to kill everything much faster than they would have killing things one at a time.
In Gothic III, you will often encounter packs of animals in the wilderness, from large birds to wolves, rhinos, and other beasts. Without exception, as soon as you attack a member of the pack, the entire group will go into berserk attack mode and chase you to the ends of the earth until either you or they are dead. One upside of this is that you can lead them into populated areas like cities, where the guards will assist you in taking them down.
Grim Dawn: Nemesis monsters act like this. Unlike any other creature and boss, which will eventually head back to their spot once you've ran enough, Nemesis monsters will always know where you are once they've seen you and will never stop chasing you until you leave to a load-in area or use a riftgate to escape; they'll even chase you right into towns like this. Somewhat justified in their origins, as a Nemesis only spawns once its respective faction hates your guts because you've killed so many of them (even if it gets odd with a faction like the Beasts, which despite being barely sentient if at all still send a titanic half-wooden chupacabra after you), so like the name indicates they're out for retribution against you specifically, likely sent by their faction directly to hunt you down.
Throughout the hovercraft chapter of Half-Life 2 the player is attacked repeated by Hunter-Choppers, but the fact that though the player gets a chance to shoot one down during the final confrontation at the dam the others just fly off for repairs, and the rebels that mount a machine gun on your hovercraft only mentioning a singular Hunter-Chopper, it's implied that it's the exact same one hounding you throughout the entire chapter and only getting temporarily driven off until you finally take it down for good at the dam.
Illidan can very easily be this in Heroes of the Storm. He moves fairly fast to begin with, and he can use his Sweeping Strike and Dive abilities to catch up to a fleeing hero. And in case that wasn't enough, he can also take The Hunt, so that if you do manage to escape his chase but then pass back into his team's line of sight, he can dive on you from across the map, bellowing "You! Are not! PREPARED!"
Most of the time, people don't do this, at least in higher levels of play. Overextending while chasing a wounded enemy hero is a very good way to get isolated from your team and killed, meaning you feed the enemy team experience and cost your side a player while you wait to respawn, for no benefit.
In Telltale's Jurassic Park video game, the T. rex will constantly attack the protagonists throughout the game despite killing enough dinosaurs over the course of it that it REALLY has no reason to. The only thing that can distract it from chasing them is a Triceratops trying to kill it, and even then, it still takes a moment to divert its attention from the Triceratops to attack the humans.
In the Sega GenesisJurassic Park game, you can play as a Velociraptor whose sole motivation is to eat Dr. Grant. Each level the raptor is following his scent, going through the same levels you'd go through playing as Dr. Grant. She could eat the hundreds of dinosaurs and armed guards she kills trying to get to Grant, but no. Maybe the raptor has a thing against paleontologists (though there is mentioned in the narrative that the Raptor is trying to follow Grant to get off the island, rather than kill him).
While Peter Jackson's version of King Kong (2005) uses this trope full-force in the movie as seen above, and you still get chased way too far by the V. rexes, the Licensed Game actually averts this as a specific game mechanic. A lot of time, you can divert the attention of a predator away from you by killing something smaller, causing the larger enemy to take the easier meal. There are even giant dragonflies and grubs you can stab with a spear, specifically for creating such distractions.
The resident Boss in Mook Clothing, The Eliminator, from Kingdom Hearts coded. In the game, is it the only enemy that not only teleports to keep you within its attack range, but after a certain level it will teleport between sector rooms. This is rather unfortunate considering its power level.
Several of the more mobile champions in League of Legends count as this, but Hecarim is the one of best. All of his abilities can be cast while moving, one greatly increases movement speed until a basic attack on an anemy, at which point he will launch both champions in his last move direction. His ultimate allows him to dravel through walls, trees, et cetera. You will never escape a Hecarim unless your team helps you. Warwick can sense low-health enemies anywhere on the map and gains a large movement speed boost while tracking them. And his main combat gimmick is being more lethal to enemies already injured, as well. Knowing when it is and isn't safe to recall to safety is a skill quickly acquired playing League.
Averted in The Legend of Zelda: Breath of the Wild by non-monstrous predatory animals such as wolves and bears; while they'll attack Link opportunistically if he stumbles upon them, they'll immediately flee upon being wounded or having a member of their group killed. In the case of the pack hunters, they'll also generally run away if Link happens upon any that are alone.
Limbo has the game's only boss, a giant spider. It has only one leg the last time it shows up, but despite that it still tries to kill you.
The Long Dark features a game mode where an unkillableold bear attacks you at the very start of the game and continually chases you throughout the map to try and kill you anytime you're outside. Shooting the bear with a flaregun will cause it to momentarily retreat (while a flare to the face is invariably fatal for other animals including normal bears). Once you get to the hunter's cabin in Mystery Lake, you can find a hunting rifle capable of hurting it and return the favour.
Impact Man from Mega Man 11 doesn't even wait for Mega Man to enter his boss room, instead breaking into his component pieces and going out to harass him in the level itself throughout several rooms.
Zig-Zagging Trope in Metro 2033. The Moscow mutants will go to great lengths for a meal, even going so far as to attack armored humans with automatic weapons. Similarly, they are not deterred by gunfire, bright light, or the smell of their own dead, as most normal animals. On the other hand, shooting them will usually cause them to back off momentarily, they do eventually flee if enough of their pack is killed, they are frightened of the anomalies and ghosts that infest the haunted tunnels, and they will also avoid large fires (though they will brave them if there is no other option). Totally averted with some other mutants — Librarians, the possibly-once-human, possibly sentient mutants that infest the Lenin Library and the Military Archives beneath it are simply territorial, and will even back down if you look them right in the eye (in the novel, one even tells Artyom to "go away" in Russian), and the Biomass under D6 ignores you until you start actively trying to kill it. The Demons, winged mutations of the tigers in the Moscow Zoo, will not try to follow you inside, though they will opportunistically try to snatch you if you spend too long outside or near windows. Overall, the unusual persistence of the Moscow predators is justified by the scarcity of food in the Metro — they are just as starved as the humans they feed on, and they really can't afford to ignore such a tempting target as an adult human (or better yet, woman and children he's guarding) unless they have no other option.
Last Light expands a bit on the situation, giving some of the mutants additional motivations for their behaviour and pushing the game closer to a full aversion of the trope. Watchmen are shown to possess a limited capacity for restraint and rational thought, being content to leave Artyom alone so long as they're not starving or if he doesn't provoke them. They are also more apt to break off an attack if they lose too many of their own, and the Baby Dark One points out that they fear what they don't understand when lending Artyom his psychic facilities to seek them out in cover with. On the other hand, Demons and the new Bear boss-mutant are both shown at various points in the game to be simply protecting their young, and the same is implied for the Nosalis Rhino (a Nosalis brood mother who watches over a large pack of them). The aquatic Shrimp are docile unless bothered by disturbances in the water, which a group of Reds promptly create while carelessly boating through Shrimp-infested tunnels.
Metroid Fusion has the SA-X, Samus' Power Suit taken over by an X-parasite which chases you through the entire game with no way to stop it. (The SA-X is fairly easy to fool, though, not being able to find Samus should she slip out of immediate view.) On the other hand, since Metroids eat X-parasites, Samus effectively plays the role of Super Persistent Predator in the game itself. Although technically Samus is also a Metroid in this game. And the SA-X has some of Samus's instincts, such as eliminating all Metroids.
The player in Miasmata is being stalked by a deadly creature that will follow him around the entire island and is liable to attack at any hour of day or night.
In the Chimera Laboratory in Mother 3, the Ultimate Chimera is chasing you around. It can't be hurt, and if you touch it you'll get an instant game over without even entering battle. What's more, in New Pork City later on, you find it an inch from your face, SITTING ON A TOILET in a stall you just opened in the bathroom dungeon.
The series at large inverts this. Most large monsters tend to flee from the player after exhausting themselves or sustaining large amounts of damage, and usually have to be chased in order to be taken down. In other words, the player is an example. Also, when hungry, monsters will leave to find easier game, generally going for the easiest to kill herbivores available or even scavenging, than the heavily armed human in front of them. They will also revert to a neutral state if you leave an area to heal up. All in all, monsters really want nothing to do with the hunter at all and will make any effort to scare him/her away or flee themselves. While this can seem quite sad, really, other material makes it clear large monsters often run because the hunter's a tough nut to crack. Plenty of quest descriptions mention stealing livestock or attacking exposed caravans, and Monster Hunter: Stories in particular shows these incidents can very easily have a body count.
Max0r: After a long and arduous battle, you (the monster) run away as far as you can. But, every time you turn around, he's still there. And even worse, IT'S WEARING YOUR SKIN.
Averted to another extent during Rampage Defence quests in Monster Hunter: Rise. Most of the waves of large monsters are merely following the tide and running into a sufficiently dense defensive line will make them go Screw This, I'm Outta Here with even less injury than in the field. Only the "Apex" monster directing the horde will stand and fight, and they're sufficiently agitated to fight to the death.
A notable exception to this is the Deviljho from Monster Hunter 3 (Tri) and onward; it is an invasive Apex Predator that will actively chase prey down to sate its never-ending hunger, no matter where it goes. This is because it has a Hyper Active Metabolism, which makes it burn energy really quickly, and allows it to invade any ecosystem. In Monster Hunter: World, the Savage Deviljho variant will even refuse to ever run from battle no matter how much damage is done to it, and will chase the player to the ends of the earth if they try to run (most monsters will give up after a certain distance).
The polar bears in Never Alone. Justified given that they're polar bears, which really will stalk humans for miles due to how scarce food is in the arctic.
There was a glitch in The Oregon Trail III that sometimes made wild animals act like this. If you went out hunting and fled from an angry bear, the bear would appear right in your face the next time you went hunting, ready to maul your party members. This bear would follow your wagon for thousands of miles until you managed to kill it.
Pikmin: The Spotty Bulbears are already Demonic Spiders on their own, but Pikmin 2 upgrades them in numerous ways, one of which is that, while every other enemy in the game (except the similarly persistent but less threatening Gatling Groinks) has a set radius they'll stay in before giving up chasing you, they have no such limit and will follow you to the ends of the earth until one of you is dead. Even then, the Bulbear's corpse will have to be harvested, or it will come back to life.
In Planet Explorers, some aggressive animals will chase the player for hours, even when mortally wounded.
Krookodile, a crook crocodilian that's said to never allow its prey to escape. Oddly, it doesn't learn the move Pursuit through normal means, though it's still better than Garchomp, who's said to be similarly persistent but doesn't learn the move at all.
While never specified to be a predator, Primeape fits the persistent part because it never gives up chasing whoever angers it till it has beaten them up. This happens a lot, because the Pokémon has such a extreme Hair-Trigger Temper it's near constantly in a Unstoppable Rage. Possibly averted in Sun and Moon, where a Pokédex entry notes that it has been known to become so angry that it dies.
In gameplay, running away from a wild Pokemon is determined primarily by the difference in speed between the combatants. Fast wild Pokemon will simply not let you escape from battle. Some of them even use a move (Mean Look) or an inherent Ability (Magnet Pull, Arena Trap) to lock you into the battle until either your 'Mon has fainted, or they have been knocked out or caught.
Quest for Glory III has Dinosaurs. Throughout the series it's possible to escape monsters simply by running for it. Not Dinosaurs. Unless you cheat by jacking the game speed up they will continue following you across the Savannah no matter how far you run.
The shark in Raft will hunt you and your vessel endlessly. You can distract it with bait, but once the bait is gone, it will come back for you. You can kill it (and butcher it for delicious shark steaks), but an identical shark will show up a day or two later to resume the hunt. Even repeatedly stabbing it with a spear or arrows doesn't convince it to back off for a moment.
In Ravensword: Shadowlands, each and every enemy, once they spot you, will not give up trying to reach and kill you until either you or them are dead, or until you completely leave the worldspace that they (or the dungeon they live in) are situated. And since you cannot use Fast Travel while someone is attacking you, it's entirely possible to get yourself stuck inside a dungeon with aggro'ed enemies waiting outside and with no ability to fight back. Add in the fact that the enemies are capable of stun-locking you if enough of them are beating you at the same time, and the fact that the game has limited saves, and you have a recipe for potentially making the game Unwinnable.
In Red Dead Redemption, cougars, wolves, and bears will chase John for ridiculous distances in which most real life predators would have long since given up and gone after easier prey. Even shooting them doesn't mean they will stop. Also, coyotes, wolves and even bobcats will attack you, when all of these animals mostly avoid humans in real life.
Nemesis from Resident Evil 3: Nemesis deserves special mention as he can follow you even to other rooms. Normally all one had to do to run away from an enemy is go into another room, but you have to run away quite a bit from Nemesis before he finally gives up. It was explicitly engineered and programmed to hunt down S.T.A.R.S. members, which Jill is. That's all it was made to do.
The T-00 (Mr. X) in Resident Evil 2 was the first draft for Nemesis, as he would follow either Leon or Claire in their B scenario throughout the entire game, but only in scripted encounters. And it's only Leon or Claire, as while he's ostensibly meant to hunt down the G-virus, in a couple of spots, he will ignore the character who has a sample of the G-virus on them (either Ada or Sherry) to chase Leon or Claire in the elevator security room. The remake, on the other hand, turns off the scripted encounters, and turns him into a relentless pursuer who will not stop until every living person in Raccoon City is dead.
Resident Evil Village continues the trend with Lady Dimitrescu, who will pursue Ethan Winters throughout her castle until her boss battle.
Spelling Jungle: Tigers (in Spelling Jungle) and wolves (in Spelling Blizzard) will chase after Wali, and won't stop unless he crosses a type of terrain that they can't. The Abominable Snowman is the same way in the second game, with the added feature of not being halted by snow like the wolves.
String Tyrant Has an enemy called The Stranger, which actively looks for Mary instead of patrolling, and respawns stronger upon defeat. It even sort of resembles Mr. X.
Invoked for horror in Sunless Sea: Most zee-monsters behave relatively normally in this regard, and will stop chasing you once it's clear you've gotten away. The Constant Companion, however, doesn't. If you dare dive beneath the surface with your Terror above 70, this gigantic arachnid monstrosity will pop itself out of the zeefloor and will not stop hunting you, even if it means having to literally chase you through the entire map. At least it can't float, so resurfacing will keep you safe, but it will gladly wait for you to come back with high Terror to begin its chase again.
Hilariously, the Green Demon Challenge of Super Mario 64 turns a 1-Up Mushroom into this. The challenge involves triggering said mushroom that homes in on Mario and then avoiding it long enough to collect the red coins. Every step of the way that 1-Up will be staring doll-eyed at the screen mercilessly chasing Mario throughout the level and it absolutely will not stop, EVER, until you get a 1up.
Part of what gives Hammer Haunts their Demonic Spider status in Thief is that, while normal guards will stop chasing you after a while and run away if their health gets too low, Hammer Haunts do neither. If they spot you, they will chase you to the ends of the earth and only death (or climbing up somewhere they can't reach) will stop them.
Justified in Vampire: The Masquerade - Bloodlines with the werewolves. Nines specifically decides to hide out in their territory because they're known to hate vampires, and no one will look for him there. Even then, they leave him alone, until a different vampire sets fire to their territory — then two of them go looking for retribution.
The Wolf and the Waves: If a zombie spots you, it will chase you across the island until it dies, even if you move far beyond its field of vision.
Usually, monsters in World of Warcraft are leashed, meaning they'll reset to previous positions if you outrun them for long enough. However, for a while after they were put into the game, there was a bug with the giant spiders of Deadwind Pass where, once aggro'd, they would never reset. They would follow you across multiple zones, through aggro reset mechanics, even through death. Even if you somehow managed to get on a flight path to get away, if you ever came back... it would be waiting.
Wurm Online: Played extremely straight with any hostile NPC animals. If you're not fast enough to get away from their aggro range, they will hunt you from one side of the map to the other and back again.
