There are many examples that make him the very personification of this trope. A similar situation and dialogue occurs in The Thing: Last Line Of Defense, which takes place when Sue Richards is pregnant with her and Reed's first kid, Franklin. Bad guy Blastaar beats Ben Grimm to a pulp, and proceeds to attack Sue. She's unconscious and is about to be killed when the half-dead Grimm gets back in the fight with:
The Thing: If that girl or her baby is hurt, you won't be leavin' this room alive. Blastaar: Bold talk from a man who can barely stand. And you still make threats. The Thing: Not a threat. A promise.
The Thing: Hold it! This fight ainít over yet... not by a long shot! Ya only won on a technicality! Ya didnít really beat me! Yaíll never beat me! Iím just too stupidÖ and uglyÖ ta know when to quit!
The Champion: No, I could never defeat you. I could crush your bones and break your body, but I could never break your spirit.
Also, say what you like about Doctor Doom, but his obsessive, all-consuming hatred of Reed Richards never, ever, ever dies, and he will keep living, and fighting, and plotting Richards' ultimate and final defeat until one of them is dead. And even then he will not give up.
Hell in Marvel Zombies where every single superpowered character succumbs to the zombie hunger. Doctor Doom is the ONLY ONE to resist it! even when CAPTAIN AMERICA, THOR and SPIDER-MAN, heroes who are renown for their Heroic Willpower give in Dr. Doom doesn't.
Batman in most incarnations. Lampshaded in an issue of Superman/Batman where the two heroes are standing in the Mall in Washington D.C., surrounded by an obscene number of supervillains.
Superman: They're all around us, you know. Batman: Do you think we can take them? I think we can take them. Superman: You always think we can take them. (Spoiler alert: turns out they can take them.)
All of the Batfamily probably counts, but Jason Todd gets a special mention. In one issue, he's hanging far above the ground, hunted by a tentacled monster as Gotham burns around them.
Tentacle Monster: Is that your super-power, boy? Too stupid to ever give up?
Jason: heh ... Maybe it is.
In Batman: The Cult, he also argues with Batman when Batman, having been tortured and had his will broken by the leader of a cult, wants to give up on Gotham, and he's the one who finally persuades him to go back on his decision. At the end of the series, he also saves Batman's life after dragging himself through the sewers with a bullet in his leg.
Stephanie Brown (Batgirl IV), in spades. Her dad was the third-rate villain the Cluemaster, her mother was an addict, she took up crime-fighting to get back at her father, and she basically spent the next several years being strongly discouraged and occasionally tolerated by Batman. Then he made her Robin, then he fired her, and then she was tortured half to death. So she faked her death which added to Tim Drake's angst, but later came back to Gotham City, safe and sound, and eventually wound up as the current Batgirl. She is currently kicking ass under the full approval of Batman.
Nightwing has the nasty habit of not letting his wounds heal. Alfred once told Oracle he would understand if she had to pump him full of tranquilizers to keep him still. Alfred has also threatened to knee-cap Nightwing's good leg to keep him for running back into battle with a leg injury.
Out of Batman's rogues gallery, Bane is probably the supreme example of this trope. His backstory includes growing up in a prison right from his birth. For several years he was held to solitary confinement, which there meant a barred off pit where all he could eat was rats, and the tides would cause it to flood twice a day. Instead of going insane or dying, it only made him stronger. The warden tried to get rid of Bane by putting him up as a test subject of an experimental drug called Venom, which had killed all the previous subjects. It made him stronger. Once Batman manages to put Bane into prison he decides that reliance on Venom has made him soft, and he kills another inmate because he wants to be sent to solitary so he can purge his addiction by going cold turkey and train to build his body back up without distractions. It worked exactly as planned.
Superman. In Emperor Joker, Batman is broken by having himself killed, night after night after night. Only Superman's selfless offer to take those memories (thanks to Mxyzptlk and the Spectre) enables him to go on. Superman is also seen in the John Byrne reboot boosting the willpower of a group of Green Lanterns' rings. He beat Doomsday after the JLA had been crushed, the story frequently describing him as never giving up.
Superman's Determinator status is best encapsulated in the following quote from "What's So Funny About Truth, Justice, and the American Way?", in which he handily defeats a cynical Nineties Antihero and declares that "Dreams save us. Dreams lift us up and transform us. And on my soul I swear... until my dream of a world where dignity, honor and justice becomes the reality we all share... Iíll never stop fighting. Ever."
Iron Man. His whole story is about determination. After being injured and kidnapped by terrorists, Tony Stark survives his injuries with the help of Yinsen to invent a powered suit of armor that would help him escape...all while under the terrorists' noses. Since then, Stark has survived shrapnel to the heart, faced numerous enemies that threaten himself and/or the world, stopped countless attempts to steal his technology, confronted his troubles with alcoholism, and whenever beaten, will do whatever upgrades it takes to come out on top again. It's no mistake that his latest company is called "Stark Resilient."
Not to mention his approach to Locking MacGyver in the Store Cupboard. No matter how unbeatable the odds are, no matter how hopeless the situation, no matter how difficult the challenge, he will find some way to figure out how to defeat the villain/save people in danger/escape/survive/put a dent in the opposition.
The Mighty Thor. Even when reduced to a pile of flesh from slaying a monster prophesied to kill him, he was able to control The Destroyer and threaten Hela into giving him his body back.
The Mask: Walter. He never stops, and appears to be the only being on the planet the mask Just. Can. Not. Kill. (and not for lack of trying, either).
Basically every main character in Sin City: Hartigan, Marv, Dwight, Wallace... some of the predicaments that these characters find themselves in are flabbergasting, yet they never show more than the slightest notion of fear in their endeavors.
The very first Sin City yarn is a prime example of this trope. A Hooker with a Heart of Gold named Goldie was killed, and Marv survived multiple injuries (he had to get himself bandaged up at least twice), including gunshots, being hit by a car and being hit in the head with a sledgehammer, to avenge her death because she slept with him (and took his virginity), even though in all likelihood she was just using him for protection and felt nothing for him (though her twin sister Wendy, Marv's companion, did come to care for him). Even when he's on Death Row and they execute him at the electric chair, it takes two tries to finish him.
"Is that the best you can do, you pansies?"
The graphic novel 300 (and really, the real life battle that it it based on) makes a note of showing that all of the 300 Spartans (particularly King Leonidas) are some of the most badassdeterminators imaginable.
It's also important to note, in final day of the real life battle, the Thespians (from Thermopylae not the other kind) who also stayed with the Spartans to the end to protect the allied retreat. The Thebians, on the other hand... not so much (they eventually surrendered).
The to be-Saint of Killers from Preacher managed to retain his mind even in death by pure hatred of the two men who he had swore vengeance on. And then he became invincible, and unstoppable, and the world trembled at the thunder of his guns.
In the Marvel G.I. Joe continuity, Snake-Eyes has just been in a helicopter crash, his crush is badly injured, and his head and throat have been burned and implanted with shrapnel-like shards of glass. Ordered to stand down, he writers two letters in the sand, with his blood: C M. Continue Mission. Then, he all but single-handedly pulls the mission off.
Or this time, where, in fact, all he could do was provide moral support:
Elsie Dee: The detonation pwogwam is too compwex to overwide! There's nothing you can do! Wolverine: You're wrong, lil' darlin'. There's something I can do real good. The real thing I do best ... I don't give up. Ever. I'm expectin' you not to give up, either.
Captain America defeats superpowered opponents with both his incredible fighting skills... and the fact that he just won't quit!
Cap: This nation was founded on one principle above all else: the requirement that we stand up for what we believe, no matter the odds or the consequences. When the mob and the press and the whole world tell you to move, your job is to plant yourself like a tree beside the river of truth, and tell the whole world — "No, you move."
One of the best examples is in Avengers #335. The genocidal Brethren have invaded Earth, hoping the legendary planet that held off Galactus can give them a good fight. Unfortunately, they're so powerful that even the Avengers are hopelessly outmatched. Cap is no exception, but he still takes on their leader, Thane Ector... and just keeps on coming at him. He loses the fight badly, but wins Ector's respect, which later proves essential.
In Jim Starlin's Infinity Gauntlet miniseries, Cap refused to surrender to an omnipowered Thanos, even when the Mad Titan had already defeated all the other heroes who were opposing him.
And Spider-Man could be his protege. That's almost as much a part of the character as Spider-Sense and webslinging.
Protege? Spidey surpasses Captain America in Determination and never calling it quits. There is not a single major villain in his roster that isn't at least half again as powerful as Spidey. This is what he does.
For one of the best examples, check out Amazing Spider-Man #229-230, where Spidey tries to stop the Juggernaut, and succeeds by steering him into a recently poured foundation of wet cement, where his own weight pulls him under like quicksand! Oh, the Juggernaut eventually breaks free, but not until long after the battle is over. Crowning Moment Of Awesome, indeed!.
Or for that matter, Amazing Spider-Man #270, where Spider-Man stops Firelord, leaving the full roster of active Avengers that were riding in as the cavalry to stand around with their mouths agape. When Captain America is impressed, you are officially impressive.
Once he kept A WHOLE, TALL BUILDING from falling, on his shoulders, allowing everyone to escape. Should be noted that, despite him having super-strength, that's WAY beyond what his powers should allow him, so it almost killed him. And he did it WHILE INJURED.
Another time, he single-handedly defeated Thermite, Blitz, Eel, Vanisher, Plantman, Tangle, the Super-Adaptoid, Dreadnought and Dragon Man. It's impressive if you consider that just between the Super-Adaptoid, Dreadnought or Dragon Man, they usually required a whole team of Avengers to be stopped.
There's also a classic issue where Spider-Man is trapped under a huge machine under piles of concrete in an underwater now-destroyed base that belonged to Dr. Octopus. The weight is truly above his superhuman strength, but Spidey still manages to escape it and proceed to the surface. Why? Because he had just recovered the only radioactive phlebotinum that could save Aunt May. Oh, and he did it after taking a beating and breaking an arm, too.
Spidey fits this trope to a 'T'. The classic two-issue "Nothing Can Stop the Juggernaut!" leaves one wondering who it was that really couldn't be stopped.
After an exhausting several-hour fight with a fellow Determinator, Morlun, Spidey eventually stops him by injecting himself with a near lethal amount of radiation, a deadly energy for the life-draining villain.
In the story arc "Revelations", which features the end of the infamous Clone Saga and the return of Norman Osborn, the following exchange takes place between Osborn and Parker:
Spider-Man: Because then, Norman, you would win. And I will never give you the satisfaction.
In Spider-Man: The Other, thanks to some mystical Spider-virus, Spider-Man is at his deathbed. Weakened, he still does his best to take on Morlun, who promptly proceeds to beat the ever-living daylight out of the guy. He leaves, and Spider-Man is rushed to the hospital, with his face so badly bruised that his own wife couldn't recognize him. Morlun shows up, and when his wife defends him, he promptly breaks her arm and comments on how he might eat her as a snack before he moves onto the main course: Spider-Man. Spidey, bruised, beaten and barely conscious, overhears this, and promptly gets out of his deathbed, beats Morlun into the ground, and stabs him through the heart. You do not mess with Spider-Man's family.
Ever so nicely summed up in this quote: "...he does it and he keeps his ideals doing it and he keeps fighting when any sane man would just lay down and die."
Also the very first Spider-Man Annual which introduced the Sinister Six where Spider-Man lost his powers and then still went to face the six most dangerous villains he had ever fought one after another because it was his responsibility. He got his powers back after the first fight.
In "The Death of Spider-Man" story line, Spidey not only saves Captain Fucking America from a sniper's bullet by taking the bullet for Cap, he also manages to find the strength to fight off the Sinister Six make that "Five"..., defeating them all. Yes, Peter ultimately succumbs to his gunshot wounds, but that's the mother of all Spider-Man Determinator moments.
The Green Lantern Corps is full of Determinators, with Hal Jordan being a big one. Seeing as their rings are fueled by will power, this is probably a given.
Hal Jordan is not the biggest Determinator among the Lanterns, just (possibly) the best of the Lanterns. Guy Gardner, for all his faults, smugness, and jerkassery, is said to have a willpower so strong that the Green Lantern rings are actually unable to contain it. He has used a red, yellow, and green ring in the past, and his will is so strong that his rings constantly emit power—even if he's just standing still, doing nothing whatsoever, his ring will spark and flash. That's right, folks. Guy Gardner's willpower is actually too strong. Which is awesome.
John Stewart also deserves a special mention. His willpower is so great, it overloaded his ring. In one issue, he tried to re-create an entire planet and would have succeeded if his ring hadn't crapped out on him due to overloading.
Kyle Rayner, without his ring, only holding an iron bar, refused to give up when he faced a fully powered Hal/Parallax, who took Kyle's ring previously. When Hal asked him why, Kyle answered that he was a hero and heroes fight for what's right, no matter how hard the situation is. Touched by Kyle's words, Hal returns him the ring and flies away.
In the Marvel G1Transformers comics, Grimlock's early hot-headed temper and stoneheaded stubbornness were revealed to be evidence of a greater quality: that of being unwilling or unable to consider giving up. He engaged in forbidden science to bring back his troops, attacked Unicron willingly, and even after the Decepticons had cut them down to five, as he explained to Prowl, post-eviscerating a Decepticon ambush: "Thatwhat we do, Prowl. We fight."
Zone from Special Forces, a severe autistic cajoled into the military by a desperate recruiting officer. He follows a list for his daily activities, and if it's on the list, it gets done. When his squad's first mission goes sour and only he and Felony are alive, he adamantly refuses to stay down; the mission was on the list, and he's not going to stop until it's completed or he's dead. Later, when he's captured, the enemy leader remarks that "Long have I heard tales of the indomitable will of the American fighting man... this one is something else. Superhuman! It's as if he could not feel pain!" Zone would eventually ride a motorcycle up a giant sword to jump into a helicopter carrying his quarry, and sacrifice himself at Felony's behest for the sake of the mission.
In the seventh issue of the original Daredevil series (wherein he changes from the yellow costume to his more familiar all-red ensemble), Daredevil alone faces off against the Sub-Mariner in a desperate bid to keep him from demolishing Manhattan in a fit of rage. Broken and beaten to a pulp, he still tries to stand and challenge Namor; this feat of courage and determination is so moving to him that he immediately leaves for Atlantis out of respect. Hard. Core. Namor even remarks that he has faced off against many very powerful enemies, from the Fantastic Four to the Avengers, but never had he seen such bravery, and from the most vulnerable challenger of all.
In the aptly-named "Hardcore" arc, Daredevil gets burned alive by Typhoid Mary (luckily for him Luke Cage is in the neighborhood). BUT, later that night, he fights off Bullseye, and then goes directly to stomp the Kingpin's ass!
Followed up by taking on some 200-odd Yakuza thugs in "King of Hells Kitchen." As Matt Murdock, presumably with just a regular cane (as he's retired from being Daredevil at the time). He's badly injured in the process, but he gives as good as he gets until the FBI provides a distraction for him to escape.
Friday in Rogue Trooper, when he crosses two hundred miles in a few days to take down Highsight while being occasionally attacked by rogue grunts.
The original Rogue tracks the Traitor General across Nu Earth to avenge his comrades.
Red has Paul Moses, although this is more because despite the fact that he's an old man and retired, several teams of professional killers sent after him and an entire building full of soldiers he has to fight through do not deter him in the slightest. They send wave after wave of men at him and he slaughters all of them without so much as a scratch.
Judge Dredd. With no means of transportation left to him, he travels through Death Valley on foot to deliver a plague vaccine to a desperate Mega City Two ("The Cursed Earth"). With Mega City One left defenseless and completely open to saturation bombing and invasion, he refuses to surrender his city to East Meg One ("Apocalypse War"). With his eyes ripped out, he still blindly runs a gauntlet against a psychic mutant monstrosity that can remodel the world around him ("City of the Damned"). There's just no stopping the guy.
In Necropolis, the Dark Judges burnt his face off, threw him into an acid river and left him for dead in the radioactive wilderness. Big mistake..
Surprisingly enough, Empowered is one if you think about it. Setting aside the fact that a sane person would have hung up the hypermembrane a long time ago rather than put up with the constant humiliation at the hands of almost everybody, her boyfriend once had to shred said suit in order to keep her from responding to an alert, concerning a local "Heavy Hitter" on the move, when she was so sick she could barely stand (then heanswered said summonswith nothing more than a sniper rifle). In later volumes, Emp demonstrates that suit or no suit, if she's mad enough, she is unstoppable. "This... is... what... I... AM!"
To be honest, it was hardly "just" a sniper rifle.
Smiley Bone from Bone shows elements of this from time to time, either because he's stupid or insanely optimistic.
His ancestor Big Johnson Bone (from the prequel Stupid, Stupid Rat Tails) lives and breathes the trope.
And possibly for the same reason as Smiley; as he seems to have a severe lack of intelligence and reason. He believes, among other things, that he has been dead before and came back to life, that he lost his sanity and found it again, caught five hundred muskrats with a single raisin, and that he encountered an Ace of Spades playing card in some extremely intricate scenario that takes a whole day just to describe and somehow involves a competition, a stuffed possum, and a massive explosion. Though, his Determinator mindset is a little more justified in that he's strong enough to kill a BEAR when he was just a newborninfant.
This is The Hulk to a T, especially when his loved ones are in danger. Regular Bruce Banner as well, he tries to save the day even when he can't turn into the Hulk. With just his brain and his wits.
The Blue Knight of Astro City, who once hunted a small-time crook over several months for the crime of unloading stolen merchandise.
The Fabulous Frog-Man is a doughy young man in a frog suit that he doesn't know how to use. But when faced with a villain, no matter how horribly outmatched he may be, he'll never stop bouncing back to give it another go. Captain America lauds this irrepressible perseverance, but also notes that this can be very bad, saying Frog-Man is the type that makes him fear another Stamford.
During the "In The Beginning" arc of The Punisher MAX, a psychotic Mafia enforcer named Pittsy quickly becomes the subject of Frank Castle's ire. After the Punisher bashes his head and injures his eye in an earlier fight, Pittsy sneaks into Castle's armory and continues their battle. Castle beats him, stabs him and gouges his eye, all the while Pittsy is screaming vulgarities and continuing to attack. Castle finally throws the gangster out a window, where he lands on and is impaled by a wrought-iron fence. He tries to pull himself off it, only to have The Punisher leap out the window and land with his entire weight on him. Later, when Castle is engaging other gangsters outside, he is shocked to see Pittsy still alive, walking with a portion of the fence still impaled through his body and still madly desiring to kill Castle. The Punisher shoots him the face with a shotgun, but the gangster still manages to walk a few more steps after that before falling. Castle tries to convince himself these last few steps were just a reflex from a dying body.
The titular Ax-CrazyHate Plague infectees in Crossed. When they set their minds to committing some sort of atrocity, nothing will deter them. In Crossed 3D, there's a recurring exchange (first seen when the protagonists watch a horde of Crossed topple a skyscraper with nothing more than torches and melee weapons) that goes like this: [character]: "How the hell did the Crossed manage to [improbable achievement]?" [other character]: "Tenacity."
The Kraken in The Umbrella Academy. Despite having the near-useless power to breathe underwater, he's one tough mother, and probably the most formidable and successful crime-fighter on the team.
Doctor Strange has been in too many situations to count where, if he had given up, he would be dead. In one memorable story he faced D'Spayre, the personification of the Despair Event Horizon, and still managed to dredge up the scrap of hope needed to defeat him.
This is -THE- essence of Donald Duck, the ultimate everyman. Life is grim, he rarely triumphs, when he does his victories tend to be hollow, no one gives him a break...and yet, he'll always keep on trying. There will be no job he won't tackle, no task he'll turn down if the thinks he can succeed at it, despite every single thing going against him.
So long as something existed to menace Mobius, Sonic from Sonic the Comic would never give up. Best exemplified by the speech he gives to Robotnik when the latter has absorbed the power of the Chaos emeralds to attain reality warping powers, while they're fighting through a series of increasingly bizarre scenarios.
Sonic: Change reality any way you like, Robotnik! Whatever you do I'll still fight you, and somehow I'll win! Because no matter how ridiculous the situation, I'll never give up, Robotnik! I'll never give up!
In one story, Sonic fights a robot that can predict his every move and counteract it. So Sonic gives up. Since the robot was programmed to believe that this circumstance would never happen, it malfunctions and blows up, which is exactly what Sonic intended.
In My Little Pony Micro Series Issue #2 Rainbow's refusal to give up is specifically mentioned by a pony reporter at the end of the comic.
Amanda Waller of DC Comics, the director of Project Cadmus and the leader of the Suicide Squad. Had the tenacity to pick up a handgun and shoot DARKSEID.
Huntress, the Helena Bertinelli version, is made of this trope. At one point, the Joker shot her three times in the stomach at point blank range, and she kept coming. Another time, she fought Lady Shiva in hand-to-hand combat, threw her own blood in Shiva's face to blind her, and managed to knock Shiva down. Shiva dubbed her the Iron Owl for this.
Wonder Woman definitely counts. Batman's plan to neutralize her during JLA: Tower of Babel was basically to trick her into believing that she was fighting an opponent equally as powerful as she was. Because she'd never give up a fight, she'd eventually drop dead from exhaustion. On a more general note, Wonder Woman possesses an incredible amount of durability and her Amazonian training gives her an immense pain tolerance. Make of that what you will.