Captain Ahab from Moby-Dick. As the principal protagonist/antagonist of a classic that's now part of the "English Canon" and being quoted and alluded to in plenty of other works since easily makes him the Trope Codifier. And Deconstructed hard. His refusal to give up or stop gets him and all but one of his crew killed.
Edmond Dantès, aka The Count of Monte Cristo, is certainly worth a mention. After his betrayal he devotes his entire life to the pursuit of vengeance. He settles down in the end, but by then he's accomplished everything he intended.
Every single Dick Francis hero/narrator character. Just to mention one: Sid Halley, who is tortured by a villain who destroys his crippled left hand, then threatens to destroy the right hand as well, the thing he most fears. Needless to say, he doesn't give up. And that's topped in a later book.
The main character of the Sword of Truth series is described at least once as "the kind of man who would jump over a cliff to come after you". Which is either Too Dumb to Live or Determinator. Or possibly both.
Hawk and Fisher, the titular characters from Simon R. Green's books, are definitely up there as determinators. Despite being completely human, they're willing to go up against anything Haven can throw at them and stick to their principles. Usually while insisting they've seen worse.
Jean Valjean's prison sentence was originally five years. It slowly gets extended to nineteen because they keep adding on time every time he tries to escape... but he won't stop trying. After he gets out (and has a nice run in with the Messiah), he's such an insatiable do-gooder that he uses disguises to keep helping the poor even when half the police in Paris are breathing down his neck.
In an inversion of the stereotypical Determinator traits, he's so relentlessly pacifist (in the book) that when the Thenardiers' gang capture him, after breaking free and securing a red-hot iron as a weapon, he brands himself with it to show them that torturing him for information that might imperil Cosette would be useless. He then discards it, even though 1) his legendary strength would have been enough to take down the lot of them, and 2) it's virtually suicide to stay there. (In the book, they were amoral cutthroats that bear no resemblance to The Keystone Kops-esque bumblers in the musical.)
Inspector Javert: Despite the fact that they live in a time where recordkeeping and communications are so poor that it's laughably easy for someone to disappear just by moving to the next town, he chases the same convict across the country for decades. He only gives up after he cannot reconcile his mission with the fact that his prey has saved his life, and is in fact a good man.
Determinators have a pretty rough time of it, with the possible exceptions of Valjean and Javert - there's Eponine and her insane devotion to Marius (who's oblivious of her), the revolutionaries who stay at the barricade even after it's obvious it's become suicide, and Fantine, who keeps working to save her daughter despite losing her teeth, hair, human dignity, health, and eventually her life.
The extinct nation of Manetheren, who took this trope to absolutely crazy extremes. The Trolloc Wars devastate the world? The Red Eagle of Manetheren flies at the forefront of every battle against the Dark One's armies. The Manetheren army receives word while still on the field of battle that a massive Trolloc army has Manetheren in its sights and there's nothing they can do in time to save their home from a horrific fate? They march home faster than even their allies thought humanly possible and meet the army before it crosses the river into their territory. Said army includes a legion of Dreadlords and Ba'alzamon himself? Doesn't faze them one bit. Their aid from other nations (their one remote chance of surviving) is cut off by betrayal by the Amyrlin Seat? They keep on fighting, only crossing the river and burning its bridges when they don't have any more troops left to fight. They finally have to evacuate their beloved mountain city and flee because the Trollocs are at the gates? Some flee, but a huge part of the non-soldier population (most of which consists of farmers and shepherds with nothing but pitchforks) takes up the slack and rides out to fight the Trollocs in a titanic final battle. Every last one of them gets massacred? The Queen, fueled by her anger over the death of her beloved husband on the field of battle, nukes the entire Trolloc army with the One Power, destroying herself and the abandoned city in the process. The few survivors of Manetheren, rather than fleeing to other lands, decide to stay and rebuild what they can. Holy. Freaking. Crap.
The region becomes known as the Two Rivers, and these survivors, while forgetting their origins, are still determinators of the highest order. "We'll survive, the Light willing. And if the Light doesn't will it, we'll still survive." They build their village on the spot their king fell. Talk about "We shall not be moved."
Perrin becomes a one after his wife is captured. He's so focused on saving her that he statedly is willing to do or sacrifice anything to get her back. At one point, he isn't particularly taken aback when, due to the influence of the Dark One, a random secretary starts puking up beetles until his body wilts and only skin and bones are left, because it has nothing to do with saving her.
In the last book, this turns out to be the key to defeating the Dark One, since he can't permanently win unless every last man has given up on goodness. Rand realizes this, and proceeds to list all of the characters who are still fighting, even after 14 books' worth of problems and a 198-page losing battle.
Also, in a Crowning Moment of Awesome, Lan prefaces his murder of one of the Forsaken by losing a duel, and replying to the enemy's triumphant cries with, "I didn't come here to win. I came here to kill you."
That particular Forsaken himself also qualifies. Nobody hates The Chosen One more than Demandred. No one (except Moridin, who's an entirely different kind of villain) comes as close to matching The Chosen One's power and skill as Demandred. And no one, not armies and not the orders of the Dark One himself, will stand between Demandred and killing aforementioned Chosen One. Well, except Lan, which just shows it takes a determinator to kill a determinator.
Lisbeth Salander from Stieg Larsson's Millennium trilogy: Do whatever you want to this 4 feet 11 inches tall girl: strand her on a bed for one full year, beat her nearly to death, rape her in the most gruesome way you can imagine, attack her in the middle of a tropical storm, send half the Scandinavian police after her, shoot her in the head then bury her, she will get back and have her revenge no matter what.
Given the large number of books, it's no surprise that Discworld has featured several.
Big Fido of Men at Arms is a tiny poodle that rose up through the ranks of the feral dog population by being a small, fast, impossible to defeat, killing machine. The narration notes that you could have sandblasted him for five minutes and what was left 'still wouldn't have given up and you'd better not turn your back on it.
Then there are Zombies, who are literally fueled by their obstinate refusal to die. Reg Shoe is probably the shining example.
Vimes gets a couple of points here too. The man managed to outlive zombies, Trolls, and Golems in an alternate universe where the Watch was wiped out by Klatch. Not to mention killing two werewolves with his bare hands, getting taken off the list of acceptable targets by the Assassins' Guild (a move he was inclined to appeal), taking history by the throat and making it cry uncle...the man is the living embodiment of this trope!
Vimes has, at this point, undergone so many near-death experiences that he and Death are literally friendly acquaintances. It doesn't slow him down any.
Death has even pointed out that when this happens, Death must have a Near-VIMES experience. He's started to bring a book and a chair, since these tend to take a while.
Also, the Luggage. Yes, its sole purpose for existing was to carry luggage around in extra-dimensional space for its owner, and it was only armed with its own lid as a mouth, a big red tongue, and the hundreds of very short, very small feet it used to move around. But it has an amazing track record, fighting across multiple continents, along the bottoms of oceans, from the end of time to the beginning of creation (in that order), fighting with the incarnation of a God of madness, and smashing through an opal shell that encased it.
Not to mention having twice followed it's owner (once for Rincewind and once for Twoflower) into a different UNIVERSE, followed Rincewind into the Dungeon Dimensions (outside any sort of universe entirely), and fought Cohen the Barbarian to a draw.
Granny Weatherwax. Once Granny Weatherwax has decided to do something, nothing will stand in her way. This is a good thing when it's "defend the kingdom" and what's in her way are The Fair Folk. It's a bad thing when it's "fly this broomstick in a straight line", and what's in her way are trees.
"If I've got a fault," she said, contriving to suggest that this was only a theoretical possibility, "it's not knowing when to turn and run. And I tend to bluff with a weak hand."
The protagonist of half of John Steakley's novel Armor joins the military in a Bug War. This leads to him being in scout armor (weaker than standard issue) in a mission gone very wrong. He's the only person to survive the mission, which, due to a processing error, means he gets sent on every single high casualty raid against the bugs. He doesn't quit. He doesn't complain. He doesn't die. He just kills. Over and over again, eventually devoting a whole mental subroutine to living through constant war.
The Deliverators from Neal Stephenson's Snow Crash will get you your pizza pie within thirty minutes or else the head of the corporation will fly down by helicopter and personally apologize for wasting your time, offering your family free tickets to Sicily at a luxury resort for compensation. Needless to say, they do not give up lightly. Of course, considering that they work for Cosa Nostra Pizza, and given their boss's original full time occupation, I would not want to cause him to be 'inconvenienced' either.
Fëanor, from The Silmarillion, who has a dying vision that the Noldor will never defeat Morgoth, and tells his sons to keep their oath to take back the Silmarils at all costs anyhow. His sons die too early or break down at the end.
And that was after he got himself - and his followers - banned from Valinor for the actions they took in pursuit of the Silmarils. The Feanorians' Oath was basically to be unfettered Determinators about getting the Silmarils back. Which they sort of do, eventually. They only really break down after learning that it was all for nothing.
Morwen from The Silmarillion and The Children of Húrin is a more benign example of this trope, but is nonetheless a Determinator. Her determinator tendencies especially come to light when she insists on going to Nargothrond to look for her son in spite of the advice and caution of others.
And Morgoth of course, who keeps fighting even after his orcs, Balrogs and dragons are defeated and he's cornered in his dungeons. Then subverted: once he realizes his armies were defeated, Morgoth begs for mercy.
Beren as well, who was determined to marry his love, even if the father sends him to an impossible quest as a condition.
Lúthien herself is equally determined (and even more successful than Beren.)
Speaking of J. R. R. Tolkien, Aragorn, Legolas, and Gimli's pursuit of the orc band who kidnapped Merry and Pippin in The Lord of the Rings shows that they have a bit of The Determinator in them as well. And then there's Samwise Gamgee, a simple gardener and loyal friend/servant/batman to Frodo.
What about Frodo and Gollum? Sam persevered because he still had hope, Frodo had no hope and yet still went doggedly on.
It was not as though Stanis thought he could get away. It was just a matter of principle: don't give up before someone fries your head with a laser, and don't do it yourself. This, after all... was the difference between him and the poor terrorist he stopped [at the spaceport by tackling him immediately upon noticing the mark of a symbiont, despite himself having broken legs from a recent botched operation and disabled prosthetics]. He did not seek death. He sought victory.
Raistlin Majere of Dragonlance was willing to sacrifice anything for his goal - his brother, his love, his health, his sanity, his life. By the way, his goal? To defeat and replace the gods.
Inigo Montoya from The Princess Bride, once he had his target in sight. His "overdeveloped sense of vengeance" drove him onward to a truly iconicCrowning Moment Of Awesome. Yet he was an amateur compared to Westley, whose modus operandi was always "Do whatever it takes, even if I have to work ten times harder than a normal person or become better than the best, and reach my goal no matter how impossible." Admittedly, even he almost gave up in face of the impossible mission of Storming the Castle, but that was before he heard they had a wheelbarrow at their disposal.
Prince Corwin, from Roger Zelazny's The Chronicles of Amber series. In the first book he fights his way up a thousand-foot staircase packed with enemy soldiers. Though he is overpowered, captured, starved, and has his eyes burnt out of his head, he eventually gets better and escapes. While still emaciated and weak he encounters an injured man threatened by monsters; he kills the monsters, builds a cairn over the dead with hundred-pound stones, then runs for a day and a night while carrying the wounded man in his arms, without pausing to rest. After that, he resolves to get back in shape.
Khalifa in Bones of the Hills pursues the Mongol expeditionary force across approximately 150 miles of desert, during which time several horses on both sides die and the Mongols' and Arabs' eyes are rubbed red by dust. Unfortunately for him, Jochi and Jebe paced themselves to always remain just outside the range of the Arabs' bows so that, when dawn broke, the Mongols could turn around and more easily pick off the slightly less tired Arabs. When this happened, Jebe ordered their Chinese conscripts to move to the back line, but Shu Ten proved his determinator credentials by begging the generals to let his men fight on the front lines despite not having the toughness or endurance that comes from growing up in the steppes.
The Star Wars Expanded Universe was aware that fans liked Boba Fett enough that he couldn't just be left to die ignominiously. So how did he escape something that preserves and digests victims over the course of millenia, trapping them in their own and each other's memories, while keeping them entirely immobile? With great difficulty. Go to And I Must Scream and ctrl-f "Sarlacc".
Supreme Commander Pellaeon is an interesting aversion. Despite being on the side infamous for a wasteful blaze-of-glory Last Stands every time they're defeated, he holds the remnants of the Empire together even as the odds get increasingly desperate. But he does so by having incredible patience, not by being the antithesis of patience.
Tom Purdom's novella "Bank Run" includes "purpose-conditioned" mercenaries, psychologically programmed to be Determinators.
Bud White spends years investigating a series of prostitute killings only he believes to be connected. When he finally gets his man and beats him to death despite suffering massive injuries in the process, "Bud White refused to die. (...) He survived massive shock, neurological trauma, the loss of over half the blood in his body."
From the Warhammer 40,000Last Chancer's novels comes Colonel Schaeffer. He and his personally chosen squads of felons and prisoners are given the most dangerous, desperate and vital missions the Inquisition can come up with. He's been run over by a tank, had his eyes cut out, shot uncounted times and gotten into fist fights with daemons. The Mechanicus has kept him alive for over 300 years by adding new parts to him whenever the old ones get shot off. To quote one of the men under his command, 'He has never failed'.
In Mossflower, Martin the Warrior fits this trope perfectly. After being repeatedly savaged by a wildcat and being knocked down time after time, this mouse keeps rising back up to fight some more, refusing to just lie down and die.
Miles Vorkosigan of the Vorkosigan Saga has this approach to pretty much anything he sets his mind to. Four foot nine inches tall, with brittle bones, he really wants to go into the army. Aged seventeen, he undergoes a Training from Hell in order to be allowed to try the physical... and breaks both his legs a few minutes in. A normal person would choose another career at this point. Miles... finds a side entrance. And later, when he asks someone to marry him... he always tries again when he gets a refusal. From three different women. In one case, repeatedly over the course of several years. Just as well the man has charm. (However, one lesson he had to learn in order to finally get his mate was that he shouldn't approach love like a military campaign.)
Bigwig from Watership Down. His Chief Rabbit told him to defend that run, and he's going to damn well do it until the Chief tells him to stop!
In the process, he sends packing in fear a rabbit who is rumoured to be Death's first cousin.
Roran of the Inheritance Cycle, a man whose determination carries him to accomplish almost ludicrously extreme feats. When his village becomes condemned by The Empire, he uses his potent charisma to convince his people to flee their homes and travel from the northern tip of Alagaesia to the sun-drenched country of Surda in the far south, avoiding Galbatorix's troops all the while. He joins the Varden, kills the Twins (two extremely powerful magicians) with his hammer, slaughters 193 enemy soldiers in one go, survives being given 50 lashes to the back by Nasuada for insubordination and is up and fighting again a few days later, journeys across Alagaesia to rescue his kidnapped girlfriend from the mountain lair Helgrind, wrestles a battle-crazed urgal to the ground until the beast surrenders and acknowledges Roran as the stronger, and rises his way up to a commander in the Varden after only a couple of months of service. And he does this all without any magic whatsoever. Yeah, Eragon doesn't look so impressive next to that, does he?
Galbatorix invoked this trope by casting spells to prevent his men from feeling pain. In the first encounter, they all died... but took out several Varden soldiers each. Still not as impressive as Roran, who actually felt pain from wounds.
Ethan Gage from William Dietrich's books Napoleon's Pyramids and The Rosetta Key. No matter what his enemies throw at him, he manages to survive it, including dangling him over a pit of snakes, burying him up to his neck in the middle of the desert and sending an entire (Napoleonic) French military brigade after him. His enemies ask him whether or not he is immortal on several occasions, Including Napoleon right before his planned execution
If Victor Cachat decides something is in the best interests of the Republic of Haven, it's going to happen, regardless of the cost to himself or others. He's the one who, to borrow from Crown of Slaves, would neither flinch, nor hesitate, nor cry in pain or fear. Not ever.
Honor herself, in the words of Earl White Haven:
It was a merciless something, her "monster" - something that went far beyond military talent, or skills, or even courage. Those things, he knew without conceit, he, too, possessed in plenty. But not that deeply personal something at the core of her, as unstoppable as Juggernaut, merciless and colder than space itself, that no sane human being would ever willingly rouse. In that instant her husband knew, with an icy shiver which somehow, perversely, only made him love her even more deeply, that as he gazed into those agate-hard eyes, he looked into the gates of Hell itself. And whatever anyone else might think, he knew now that there was no fire in Hell. There was only the handmaiden of death, and ice, and purpose, and a determination which would not - could not - relent or rest.
Treecats generally are described as having two kinds of enemies - those who have been dealt with appropriately, and those who are still alive. Given their druthers, they won't stop until the latter have become the former.
Lloyd Douglas' The Robe characterized the Jewish people as this. A disgruntled Roman soldier remarks that 'A Jew will climb out of his grave and continue to fight.' Given the repeated Jewish Rebellions he had a point, and meant it as a semi-compliment.
Ben Hanscom in IT, in his story of how he lost weight.
Also Bill Denbrough in his quest to get revenge for his brother's murder.
It sucks to be Harry Dresden. It's a good thing he's Made of Iron, since by the time the final confrontation has rolled around, he's nearly always in no condition to be walking around, much less fighting the book's Big Bad. By the end of Dead Beat, for example, he's been kicked around by Cowl, Mind Raped by Corpsetaker, had a shuriken lodged in his leg, beaten to within an inch of his life with a chain, bitten by snakes, and knocked out with a blow to the head, and the only reason he can even move is because he's blocking out the pain. He still finds the resolve to REANIMATE A TYRANNOSAURUS and fight his way through a horde of zombies, two necromancers, and a ninja ghoul.
Harry has said multiple times that he is one of the top when it comes to sheer magical capacity. What balances this out is his horrible lack of control. So it would be far easier for him to animate a T.Rex then a regular, fine-control zombie (which was forbidden anyway).
Michael sums it up best: " Because I know you, Harry Dresden. You are pathologically incapable of knowing when to quit. You don't surrender."
Miss Gard. She is quite literally disembowelled and proceeds to stuff her own guts back where they belong and seal the wound shut with superglue. Even Harry is somewhere between impressed and "Oh my God, stop that" horrified watching her do it.
And after said disembowelment, she still manages to fight off a two-thousand year old sorceress who is paired with a fallen angel.
Cowl. My God, Cowl. Let's see what he's survived...
Harry flipping a car on top of him using hellfire.
Death curses bounce off of him like ping-pong balls, without even slowing him down.
The freaking Darkhallow ritual blowing up in his face. Repeat: he survives enough magic blasted at him to ascend a mortal to a god.
Really, in the Crapsack World of the Dresden Files, everyone is a determinator at least once. From a badass Knight of the Cross, to the tiniest Wildfae. The only ones who haven't been seen determinating are gods and someone of the strongest Fae.
The Golden Oecumene often seems to be an experiment in how much can be taken away from one character while keeping it plausible that he'd remain sane. At his lowest, protagonist Phaethon has lost every single thing he's ever had, including his reputation, and fights alone against a conspiracy that everyone else believes is all in his head. Tellingly, the first time the narrative really lets up on him is when it takes another character, Atkins, and has him temporarily convinced he's Phaethon. Formerly portrayed as The Stoic, he's reduced to sobbing and begging before an artificially induced Snap Back.
Except that the sobbing and begging is because he wants to STAY Phaethon. Which invites the question: if it's that hard to be Phaethon, then, just how much worse must it be to be Atkins?
Many in The Malazan Book of the Fallen, but one of the standouts has got to be Spinnock Durav, who spends an entire night duelling against fellow Determinator and Villainous Badass GrandpaKallor. Spinnock never has a chance, yet he repeatedly gets up, despite everything Kallor does to him, all because his master asked him to ensure that the former High King never reached the city of Darujhistan. He succeeds, despite never landing a blow on Kallor, all because he's too stubborn to fall down. Kallor himself shows off his Determinator status in the same book, fighting his way through Spinnock and a freaking Dragon in his efforts to reach the city. At the rate he's going Implacable Man may not be too far off.
Bolos might fail. They might die and be destroyed. But they did not surrender, and they never — ever — quit.
This wouldn't normally be applicable to a character who just tries to survive, but Night's Eliezer just will not. Give. Up. The book shows both the good and the bad sides of this, as he becomes more willing to abandon others to save himself.
Animal Farm: Boxer will build that goddamn windmill or die trying. Of course, in the end, the latter came true.
The Railway Series: Edward, Skarloey, and Rheneas all demonstrate this trope. In chronological order:
Skarloey and Rheneas's railway is on extremely hard times. Skarloey is out of commission, leaving Rheneas to do the entire workload. Rheneas is in little better condition than his brother. One rainy day, he is traveling home with an overloaded train (there are even passengers in the Guard's van). Out on the loneliest part of the line, his valve gear on one side jams, leaving him with only one good cylinder. Neither this nor the storm-slicked rails stop him from getting the train to the station. Had he failed the railway would have closed. Instead his valor earns the Skarloey Railway praise and good publicity, beginning to turn its fortunes around.
A few years later, after Sir Handel and Peter Sam arrived and while Rheneas is away being overhauled, Peter Sam being repaired in-house and damage to Sir Handel's wheels create quite the dilemma. Skarloey volunteers to pull the train in spite of his decrepit condition. On the return journey, a spring in his frame breaks, but he manages to bring the train home. James, who had been impatiently waiting for Skarloey, swallows his irritation and leaves in respectful silence after collecting his passengers. Skarloey is also sent off for an overhaul.
Edward manages to bring a train home in spite of a broken crank pin.
Red Storm Rising: The captain of the merchant vessel Julius Fucik, which is used by the Soviets to sneak troops into Iceland, valiantly stays at his post to guide the ship into harbor even as he bleeds to death from wounds sustained from 20mm cannon rounds, refusing all but the barest minimum of medical attention until his task is finished.
Many characters in A Song of Ice and Fire fit this trope, especially Balon Greyjoy, Catelyn Stark, Daenerys, and Lord Varys. It is deconstructed in the form of Brienne of Tarth, who would love to give up her quest, return her Cool Sword, and go home to her family... but won't, because she gave her word. By the fourth book she's got some fairly major psychological trauma and her determinator characteristics may have led to her death.
Stannis Baratheon, a man unloved by even his own brothers and the smallest army in the War of Five Kings, is by far the most determined with not even murdering his own blood or devastating defeat being enough to cool his ambition of becoming the King of Westeros, because the throne is his by right. He is also the only one of the original five kings to still be alive as of book five.
Let's look at his very impressive track record for the War: Rallies the smallest army, held up mostly by sellswords and sellsails, and prepares to fight the largest army in Westeros, led by his younger, more charismatic brother, and defeating it with one death. He leads his armies, made up of Dragonstone, sellswords and the Stormlands, and attacks the capital of the continent, almost takes it, until a deus ex machina by Tywin Lannister defeats him. Undeterred, he decides to take his recently thrashed army up north, WAY up north and beat down the Wildlings, and prepares to earn the support of the North by taking down the Boltons. So, he marches his army, through a freaking blizzard, to Winterfell, to take them down once and for all.
Tywin Lannister: This is Stannis Baratheon. This man will fight to the bitter end and then some.
Keladry of Mindelan is not called "The Protector of the Small" by a handful of deities for nothing. If she has breath in her body, she will protect her people- no matter if that means risking having to re-do her long and grueling page years to save her maid, or going after a group of kidnapped children( and their 150 well armed kidnappers) with less than 20 men, some birds, a cat and a few dogs. If someone is under her protection, Kel is not going to stand around and wait for a miracle, she's gonna be that miracle or die trying, and no one's gonna talk her out of it. Her Establishing Character Moment has her as a ten-year-old girl attacking a pack of boys trying to drown some kittens, then when a giant spider monster shows up to eat the kittens, she tries to kill it with rocks.
On a lesser note, during her page training, when she was ten to fourteen years old, she was the first girl in centuries to try to become a knight without pulling a Sweet Polly Oliver. And a lot of people hated her for it; her own training master was constantly Moving the Goalposts, insinuating that she should Stay in the Kitchen, and turning a blind eye to the other pages who trying to make her quit. Said other pages conspired to make her late and thus receive punishment work, forced her to accept an overlarge and difficult mount, and gave her practice weapons that were weighted and harder to use. Kel kept going anyway, training to build muscle and negotiate with the horse and never complaining, and was stronger for it - and she made time on the side to be a Bully Hunter, too. Eventually the training master begrudgingly came to respect her and even consider her the best knight he'd ever trained, due in large part to that determination.
Beka Cooper of the City Guard is not called Terrier in her first appearance for nothing either. She WILL find the answer to the deadly mysteries that come into her beat, and despite her training partners' best attempts, she forgets when to back down.
Same writer, different series as the above, Briar Moss in his book from the Circle of Magic quartet. He will not let his mentor die, even if that means following her into the afterlife and convincing her to come back.
Deconstructed in Paradise Lost. Humanity can't return to the Garden of Eden specifically because Satan refuses to give up. Interestingly, Satan's status as a determinator has convinced a number of people that he's the actual hero of the story.
Well, by that point he was certifiably batshit. However, it was definitely his Determinator tendencies that brought him to that point: he dragged his entire family off to the Belgian Congo to be missionaries, even though the missionary agency would not approve his mission and gave them virtually no financial support, and stubbornly stayed on, doggedly trying to convert the entirely uninterested and frequently hostile locals. This all through the violence before, during, and after the Congo gained independence, through the death of his youngest daughter and being abandoned by his wife and three remaining daughters, and being driven to live alone in the jungle by the locals, who'd had just about enough of him.
Orleanna, Nathan's wife, is also the Determinator, with her fight first to keep her family alive in the Congo, then to get them the hell out. Two of her daughters, Adah and Leah, also develop into impressive Determinators. Apparently it's a family trait.
If you think about it, the Tortoise from "The Tortoise and the Hare" is this. Despite the fact that it seemed hopeless right from the start, the tortoise just keeps going and ended up winning the race.
Dicey in Homecoming. At the age of 13, leads her 10-, 9-, and 6-year-old siblings, homeless and walking much of the way, from Massachusetts to Maryland. She keeps them together, keeps them fed, and keeps them moving. To paraphrase another character later in the series, "Look it up on a map."
Time Scout: Do not mess with Skeeter Jackson's adopted family. He will spend sleepless days hunting you down through the bowels of the earth. He will hunt you down to the ends of time. He will face a storm of bullets, sprint across London, and leap a river of molten bronze. And then he'll offer you a Sadistic Choice.
Similarly, Lupus Mortiferus will get back his money. And his revenge.
Elijah Beckett from The Corsay Books is a textbook example, doggedly doing his physically-demanding and spiritually-draining duty despite debilitating illness and a growing narcotic addiction.
The Do-Gooders in Audrey, Wait! continued playing at their gig even after the ceiling collapsed. Their first offer from a label turns out to be a sham, but they push through until they get another offer from another label.
Sisterhood Series by Fern Michaels: Jack Emery and Ted Robinson become these as the series goes on. Both of them are told to their faces that the are obsessed with arresting the Vigilantes and to just give it up. Jack gives up and starts helping out the Vigilantes in The Jury. Ted gives up and starts helping out the Vigilantes in Collateral Damage.
Walker Boh of The Heritage of Shannara and The Voyage of the Jerle Shannara, takes one beating after another, but never ever gives up. Whether it's cutting off his own arm to make an escape, facing down one of the True Fae, refusing to let the Four Horsemen of the Apocalypse keep him trapped in Paranor, killing the Big Bad after having been swarmed by dozens of his minions, or surviving for half a book with a fatal wound and not only killing the Disc One Final Boss that did it to him, but redeeming one of the other villains in the process, Walker cannot be stopped. His predecessor, Allanon was also something of a determinator, but Walker takes it to new heights.
The Warlock Lord was such a determinator that he managed to cheat death by convincing himself he could not die.
Roy Merritt from Daemon has this as his defining personality trait. Video of him successfully breaking into a death-trap-filled mansion while on fire gets passed around the Darknet for years, earning him the name Burning Man. Mind you, the darknet is the network built by the system Merritt was fighting against - his determination is so impressive even his enemies are in awe.
The defining personality trait of Richard Hannay from The 39 Steps, both mentally and physically. Need someone to walk into wartime Germany, find out their secret weapon, evade capture while staggering through the snowy woods in winter with a fever, and foil the villains' evil plans long after anyone else would have given up in exhaustion or succumbed to exposure? Richard Hannay is your man.
Harry Potter: Lord Voldemort. Even with his last best lieutenant dead, Harry having refused to die yet again, his horcuxes destroyed and the Elder Wand refusing to obey him, he just keeps fighting. Pride of course, comes before the fall.
Matteo Ta'anari of Someone Else's War is the ultimate determinator. He loses his parents and his little brother, his home, his religion is compromised, he gets shot through the throat, becomes claustrophobic, sustains burns to 90% of his body, is rendered deaf-mute...and still keeps trying to free his friends from the Lord's Resistance Army. Without violence. And succeeds.
ColonelSebastianMoran of the Sherlock Holmes stories. He spent three years ruthlessly hunting Holmes all across Europe and Asia, and he's famous in India because he once crawled down a sewer drain after a wounded man-eating tiger. And won.
It goes without saying that every Victor of The Hunger Games fits this trope (with the possible exception of Annie Cresta). However it is taken up to eleven by Haymitch, Katniss and Peeta.
In the first book Haymitch consciously uses this trope during the Games. He knows Katniss is determined to win no matter what and that Peeta is just as determined to keep her alive. Haymitch figures that between the three of them they can get Katniss home so he focuses all his efforts on helping her survive while ignoring Peeta until he becomes useful for that purpose. Peeta... doesn't mind one bit since he wants Katniss to be the one who survives.
In Catching Fire it becomes problematic rather than helpful. Katniss and Peeta are both as determined as ever only now they're at cross purposes. Peeta still wants to keep Katniss alive but she's no longer focused on her own survival since she wants Peeta to be the one who lives.
Even more complicated in Mockingjay. Having been hijacked by the Capitol Peeta's incredible determination is now focused on killing Katniss. A more subtle example over the course of the second half of the book but the fact that he manages to overcome the effects of the Capitol's torture and apparently being the only person ever to recover from hijacking speaks volumes about his determination.
Include Annie in this trope as well. She had already snapped, she could have just given up and let herself drown when they flooded the arena, but she swam and swam until everyone else was dead and she won. Maybe she wasn't even thinking about it, but whatever survival instinct she was running on led her to outlast the others.
Musashi is this. Nothing will stop him from pursuing the Way of the Sword. One character comments that it's his determination, more than anything else, that is the secret to his success.
Peter Watson, the child protagonist of Roald Dahl's short story "The Swan" (collected in The Wonderful Story of Henry Sugar and Six More), is described thusly in the closing moments of the story:
"Some people, when they have taken too much and have been driven beyond the point of endurance, simply crumple and give up. There are others, though they are not many, who will for some reason always be unconquerable. You meet them in time of war and also in time of peace. They have an indomitable spirit and nothing, neither pain nor torture nor threat of death, will cause them to give up."
And how does he earn that description? After being bullied by two rifle-toting local youths who've murdered a mother swan, strapped its bloodied wings to the boy's arms, and forced him up a tree, and then begun opening fire on him to force him to jump to his death...he is so consumed with determination not to give in to them that he turns into a swan and flies to safety.