Arrow: Oliver Queen has sustained numerous serious injuries and been tortured many times, but has always survived through his own will.
Malcolm Merlyn also qualifies. First he almost completed The Undertaking which had effect anyway because of redundant earthquake device, then presumed dead he started rebuilding his empire. He showed up again during Mirakuru crisis to save his daughter Thea. After which he trains her and, with help from Oliver he take on Ra's'al Ghul and takes his place. Not to mention that he makes Team Arrow, many of which should hate it both because what he did to the city and because to what he did to their loved ones, if not like him, at least tolerate his presence when he is around.
Stannis Baratheon begins the war as the weakest and least popular king, but moves to attack his strongest rival anyway, refusing to parlay with anyone he considers a usurper or let even his own brother stand in his way. Even after a catastrophic defeat, he refuses to relinquish his claim. Davos says it best: "As long as Stannis lives, the war is not over."
You can remind Balon Greyjoy about the massive failure that was his first rebellion, the personal tragedies it wrought including the death of two of his sons, and how he can never hope to hold the lands he may conquer, but he'll launch a new rebellion anyway.
The Blackfish describes Catelyn's father Hoster in these terms: "I was surprised when he died. I didn't think Death had the patience."
Jorah fights this way in "The Dance of Dragons." He gets beat up in each fight, but through grit and luck manages to survive each encounter.
Dorne as a whole. Of all the Seven Kingdoms, Dorne was the last to bend the knee to the Iron Throne. Not to mention having the honor of successfully resisting Aegon Targaryen and his dragons, something that not even the fiercely independent North accomplished.
Jon Snow's skills are only one reason he keeps surviving. His endurance is the other. Jon has gone through several desperate, long-running battles against superior foes, often as the first man in and the last man standing. The Battle of the Bastards is a clear example of this, as he is able to come to the end of the fight absolutely exhausted, and still have the strength and presence of mind to outfight a fresh Ramsay Bolton. And he does this because he absolutely refuses die until the realm and the people he has sworn to protect are safe, and justice is served by his hand.
The Knick: Dr. Algernon Edwards is a highly educated surgeon, and has experience working with some of the finest doctors in Europe. None of that really matters, though, in 1900 New York City, where the only thing people see is the color of his skin. Nevertheless, he's determined to make it as a surgeon in a white man's profession and an otherwise all-white hospital, and despite the personal disdain of most of the staff. Why, you ask? Because his superior, Dr. John Thackery, is something of a mad genius and Edwards intends to learn everything he knows.
Happy Endings provides a great example of this in the third season episode, "Un-sabotagable". Chase, a guest character who once accidentally had his life ruined by Max was determined to ruin Max's life. He shows up in the beginning of the episode, says Max's life is too bad to wreck, and Max builds his life up to prove him wrong, paranoid that Chase might be helping from the shadows to knock him down later, even though Chase is not seen. It turns out at the end of the episode, Chase hit his head (again due unintentionally to Max) and fell into a coma for three days. He returned from the brink of the afterlife in order to get revenge.
Chase:But I'm so glad you're here, because I want to tell you something. I don't know where you fall on the existence of an afterlife, but I'm here to tell you there is another place. Its a spiritual realm ruled by powerful yet merciful beings. I was there Max, and I was headed for sweet eternal peace. But then I realized, I could not transcend to the next plane until I took care of the unfinished business of this world. The unfinished business of ruining your life. It was you. I came back for you. I have looked deep into the abyss of the next world, and it is beautiful. But before I make that pilgrimage, I will stop at nothing to lay waste to what you so tragically call your life."
The Hardy Boys Nancy Drew Mysteries: Frank Hardy. Once he latches onto a mystery, absolutely nothing stops him from pursuing it...not his brother Joe, not his father, especially not the supposed victim:
Mystery of the Flying Courier: Frank spots a girl who looks like someone he went to school with (someone who ran away and whom his father, Fenton, could not find). The girl denies being that person, tells him to shove off and leave her alone, tells him she neither wants nor needs his help, denies knowing him...and Frank continues to butt into her life and hound her until she finally confesses that yes, she is that girl...and still wants nothing to do with the Hardys, especially Frank...who still continues to butt into her life to help her, whether she wants it or not.
Creatures Who Came On Sunday: Everyone is convinced that the missing man just ditched his girlfriend, even Joe, and offers reasonable, likely explanations for the disappearance...except Frank, who obsesses on tracking the boyfriend down despite being warned off by the sheriff, so-called government agents, and even Sharon herself, to the point that Frank's determination is responsible for leading mob killers onto the boyfriend (who is in a witness protection program) and nearly getting himself, Joe, and Sharon killed.
But even if he doesn't have any powers, he just won't stay down.
Even death cannot stop this guy.
Chloe Sullivan. You could try to break her, use bribes, torture her, threaten her with jail and death, but nothing, I repeat, nothing is going to make her betray Clark Kent. The lengths to which the pair would go to for each other is amazing and heartwarming.
Legion: After Clark saved the world when it would be a lot easier (physically, not emotionally) to kill Chloe:
Chloe: Thanks for not giving up on me.
Clark: I'd never give up on you, Chloe. You know that.
Chloe: I just hope that if it turned out that there was no other choice, you would choose to do the right thing.
Clark: What choice would that be?
Chloe: To do what's best for the world... and not for me. Know that I'd understand.
Clark: Well, as long as I'm around, that's never gonna happen.
Chloe: I guess that's the true test of believing in someone — knowing that their lies are there to protect you. It's not really trust if you ask someone to explain themselves.
Teen Wolf: : Many of the characters have their moments, but one of the best definitely comes from Stiles.
When the Kanima paralyzes Derek and then traps them in a lap pool, Stiles keeps the much heavier Derek and himself above water for over TWO HOURS without any assistance. This wouldn't be amazing at all for the werewolves, but Stiles is human, and a pretty unathletic and scrawny one at that. Even veteran water polo players would've had extreme difficulty in that type of situation.
There's the time Stiles more-or-less made his way across the police station - which was, at the time, a BATTLEGROUND between the Argents, the Kanima, and the werewolves - in order to reach and/or rescue his father. Crawling on the ground; on his belly; with the majority of his body still paralyzed from the Kanima's venom.
The Prisoner: Number Freaking Six: "I will not make any deals with you. I resigned. I will not be pushed, filed, stamped, indexed, briefed, debriefed, or numbered! My life is my own."
All the Rangers in Mighty Morphin' Power Rangers, and many other Zordon Era Rangers. Quoth the full version of the original theme song, "No one will ever take them down, the power lies on their si-i-i-i-i-i-i-ide!" Adam gets special mention for his bravery in "Always a Chance". He morphs using his damaged original morpher and the damaged Mastodon Power Coin within. He fights and helps Carlos defeat the monster and the Quantrons in spite of the damaged powers hurting him the whole time and almostkillinghim. Rita Repulsa and Lord Zedd also count. Defeat after defeat after defeat, even forced to abandon their home palace in Zeo, but does that make them quit? Hell no.
The Daleks, utterly devoted to their self-imposed purpose of becoming the only form of life in the universe. Part of this is how they perceive themselves, but you have to admire a species that manages to survive even after being made extinct. Twice. At least. In the new series alone.
The original series' Cybermen are up there too. By the end they're a pathetic bunch of tin soldiers floating around the galaxy in a derelict spaceship, with no home planet and no influence... and they still refuse to lay down and die. "They never get tired, and they never give up."
Davros has survived several No One Could Survive That! scenarios through sheer willpower, and he has never abandoned his dream of ruling the universe through the Daleks.
While he might not look it, Rory Williams most definitely qualifies.
He watches over his fiancee's body for two thousand years until she can be brought Back from the Dead. The Doctor's astounded he didn't go insane.
And for Rory, that's just a starting point. When he really gets going, he can make a Cyberman armada back down pretty much with pure force of will. River Song obviously gets certain things from her Dad. Hell, not even death is enough to stop him. So far he's been dissolved, shot, erased from existence, burned alive, shot again, taken by a vengeful spirit, drowned, and aged to death. And he just keeps coming back.
Shows up again in 'The Wedding of River Song' in an epic manner. It's revealed that the Silence leaked the eye-drive design to humans and it's capable of electrocuting them to death, leading Amy to order Rory to remove his eye-drive, lest it happen to him. Rory casually reveals that he's actually been electrocuted for several minutes beforehand, but has kept it on and remained silent because he needed to remember what he's facing in order to protect her. However, as soon as the Silence show up, he's in too much of a pain to shoot at them. Then Amy comes back with a submachinegun.
How about the Doctor himself? If there's a wrong to be righted (and there always is), he absolutely will not stop until it is.
"No matter what happens, no matter the odds, we never, ever,ever give up!"
In the Eighth Doctor Adventures novel Seeing I the Doctor is trapped in an inescapable prison. He makes ninety-seven escape attempts in the first eighteen months.
"You've got to stop!" Akalu went on. "You've got to stop before you destroy yourself. Don't you understand? You've done enough. You have done enough. You've got to know when to stop." "Iíve got to keep going," whispered the Doctor. "I mustn't stop for anything."
"The Name of the Doctor" and "The Day of the Doctor" reveal that the Doctor considers this an inherent part of his identity. He considers his name, the Doctor, to be a promise. When asked what the promise is:
Tenth Doctor: Never cruel nor cowardly. War Doctor: Never give up. Never give in.
In Let's Kill Hitler, the Doctor is literally dying and in pain but manages to act incredibly normal (for him) at least until the last 5 or so minutes, even taking the time to change. He also focuses all his energy on saving Amy and Rory, even when he is in such pain he is unable to move.
The Doctor spends almost a thousand years, in what he believes to be his final incarnation, defending a town on the planet Christmas - and, by extension, the Time Lords - from pretty much every enemy he's ever had.
In Heaven Sent, the Doctor spends four and a half billion years trapped in an elaborate prison, repeating the same few days over and over again, with his memories resetting each time (except until the end when all his memories come flooding back) during which time he punches a hole into a 20 feet thick wall made of space diamond with nothing but his fists. It is revealed later that he could have left at any time by giving up his knowledge about a secret, but chose to stay as part of a gambit to save Clara, a companion he had fallen in love with and who had been Killed Off for Real just before he was transported into the prison/torture chamber. After he escapes the torture chamber, he proceeds to execute a successful coup against the leader of his own people in order to access time-travel tech capable of pulling Clara out of the moment of her death. Once this is complete he and Clara race literally to the last moments of the universe in a Tragic Dream of being together before the Doctor's "determinism" is forced to come to an end.
The fact that the Doctor has endured over for over 2000 years, half of that in war, and has lost almost everyone he's ever cared about but remains optimistic and compassionate is a testament to his amazing willpower.
The Master just won't stop fighting until he's dead. After Martha Jones restores the Doctor to full power in Last of the Time Lords, the Master tries to shoot him with his laser screwdriver but the blast just reverberates off the Doctor's forcefield. The Master threatens to kill his friends instead but the Doctor disarms him. The Master summons the Toclafane then teleports himself and the Doctor to Earth where he prepares to trigger his Doomsday Device but the Doctor talks him down and they return to the ship. Captain Jack destroys the Paradox Machine and the Toclafane are banished to the End of Time. The Master tries to run but Jack stops him and then the Master's wife shoots him through the heart. The Master's Pride is too great for him to surrender so he refuses to regenerate and lets himself die but is resurrected later. Then in The End of Time, he uses his lightning power to force Rassilon through the portal into the Time War and goes down with him, only to resurface in a new body in Series 8.
And of course we can't forget Rose Tyler. Neither time, space, nor alternative dimensions can stop her to get to her Doctor. The woman literally became an all powerful God just to protect her Doctor.
Martha's average day seems to take a huge amount of willpower to get through. She's a med student, which involves a huge amount of work by itself, with a highly dysfunctional family, the members of which all call her to complain about each other, which results in her having to play peace keeper. Her determinator tendencies only become more obvious once she starts travelling with the Doctor - she walked the Earth by herself for an entire year to stop the Master.
The novels also give this characterisation to the TARDIS, which strives for the entire time from the Third Doctor to the Eighth to fight a Faction Paradox biodata attack that infected the Doctor. Even destruction did not stop it; it later clawed its way back into existence as a giant anomaly known as "the Edifice" in order to continue fighting. It eventually wins, although Gallifrey is destroyed in the process.
Hidari Shotaro is impressive enough as half of Kamen Rider Double, but even as a normal human he fulfills the criteria for Badass Normal. He lets absolutely nothing stop him from protecting his client, his friends, and his partner. The most notable example is when he goes off to fight Utopia, the most powerful Dopant they've ever run into, with nothing but his own fists. And proceeds to stop Utopia's punch with his hat.
And when the Old Dopant turns him into a senior citizen, he takes his cane and goes off to kick the monster's ass. By himself. Granted he's not very effective, but points for effort.
Yup. This trope fits Mal Reynolds from Firefly to a 'T'. The man will NOT stay down. Shot in the arm? Minor annoyance. Hit in the chest with a thrown knife? Flinch and a gasp of pain. Shot in the stomach? Some medical tape and a jolt of adrenaline is all he needs. Impaled through the gut with a sword? Pull it out and keep fighting. Tortured to death? Get back up, stick the torturer with his own toy, and proceed to beat the hell out of the Big Bad.
Lose the war and every member of your platoon? Keep on flying.
Simon is a Determinator in a more quiet way. He tracked down River against all odds, continually labors to cure her and never EVER gives up.
In the episode "Safe," he gets up on the stake his sister is about to be burned at, hugs her, turns to the mob, and says "light it." No wavering for that boy.
Jack Bauer of 24, who will stop the terrorists no matter what it takes. Torturing him to death just makes him angrier. Seriously, they've tried international assassins, nuclear bombs, nerve gas, and even corrupt Presidents; and they STILL can't stop The Bauer.
G'Kar, over the course of one arc, takes off in search of a friend, and is captured by his enemy, tortured repeatedly. He only screams once the entire time (and that at Londo Mollari's silent pleading—that, and he would have died on the next whip lash had he not done so), has an eye plucked out, and breaks out of chains thought unbreakable (the chains that would have held G'Kar were weakened, but Emperor Cartagia noticed they were weak and had them replaced with stronger ones).
Sheridan. Best shown in the episode "Intersections in Real Time", where he has been captured and is put through a Nineteen Eighty-Four-esque torture session. They spend the entire episode working to break him, to get him to accept their version of the "truth", because the truth is fluid. Sheridan throws it back in the best way.
Sheridan: You know, it's funny, I was thinking about what you said, that the preeminent truth of our age is that you cannot fight the system. But if, as you say, the truth is fluid, that the truth is subjective, then maybe you can fight the system. As long as just one person refuses to be broken, refuses to bow down. Interrogator: But can you win? Sheridan: Every time I say no. Interrogator: Captain John Sheridan, will you or will you not sign a confession and endorse it before a public hearing? Sheridan:No.
Omar Little from The Wire has robbed Baltimore's most notorious drug crews for almost a decade, but in Season 5 he amps it up by taking on a whole crew out to kill him by himself, escapes an ambush by jumping out of a 6 story window and keeps on going despite having to walk with a broken leg. Even more amazingly, the person he's based on escaped the same situation by jumping from an even higher window, but this was viewed as being too unbelievable for the audience so it was tuned down.
Not with violence, but Karl "Helo" Agathon in the Battlestar Galactica episode "The Woman King". Say what you like about the quality of the episode as a whole, but character-wise, seeing that something definitely wrong is happening, he refuses to drop his investigation, over the racism of his crewmates and demands from his superiors to just let it go.
Tigh: You may as well take whatever credibility you have left and chuck it out an airlock. You seriously want to stand up for these crazy frakkin' people? What is it with you?
Starbuck: "Do what we always do. Fight 'em till we can't." In the months after she makes that statement, we see how far she takes it (specifically, to the point where she kills the same Cylon five times despite being imprisoned and deprived of anything her captor thinks could serve as a weapon, including eating implements).
Laura Roslin(after being told Adama is dead): "No. Not now. Not ever. Do you hear me? I will use every cannon, every bomb, every bullet, every weapon I have down to my own eye teeth to end you! I swear it! I'm coming for all of you!"
William Adama isn't exactly known for backing down in face of adversity...
James T. Kirk. In Star Trek II: The Wrath of Khan, it is pointed out that he is so determined to win that he cheated in his "Kobayashi Maru Test", an intentional no-win situation designed to test the character of officers-in-training. He won. The Character Development of the film is that he must deal with losing Spock; there is no way he can keep Spock from dying.
Khan from the same film. He's successfully stolen a ship, and can go anywhere he wants, but first he has to have his revenge on Kirk.
Khan: He tasks me! He tasks me and I shall have him. I'll chase him round the moons of Nibia and round the Antares malestrom and round perdition's flames before I give him up!
Q: They will follow this ship until you exhaust your fuel. They will wear down your defenses. Then you will be theirs. You can't outrun them. You can't destroy them. If you damage them, the essence of what they are remains. They regenerate and keep coming! Eventually you will weaken; your reserves will be gone. They are relentless.
The majority of the Next Generation episode The Hunted depicts the Enterprise crew trying and repeatedly failing to capture the super soldier Roga Danar, who is running loose in the Enterprise and refuses to be captured, despite the odds.
Likewise, the Jem'Hadar.
Omet'iklan: I am First Omet'iklan, and I am dead. As of this moment, we are all dead. We go into battle to reclaim our lives. This we do gladly, for we are Jem'Hadar. Remember, victory is life.
And, in the same episode
Miles O'Brien: I am Chief Miles Edward O'Brien. I am very much alive, and I intend to stay that way!
Worf, in By Inferno's Light.
First Ikat'ika: I yield. I cannot defeat this Klingon. All I can do is kill him, and that no longer holds my interest.
Garak, also in By Inferno's Light, who engineers everyone's escape by defying a chronically debilitating phobia. His determination even earns the respect of the Klingons.
General Martok: There is no greater enemy than one's own fears.
Worf: It takes a brave man to face them.
The cool ships are like this as well. Every version of the Enterprise has taken abuse that would reduce another ship to recycled scrap metal. The USS Voyager was stranded a bajillion light-years from any kind of base that could offer repairs and yet managed to take on, and defeat, hundreds of enemy ships and leave a trail of destruction and reconstruction on its way home. And of course, you have the Defiant which takes the design principles of the preceeding, turns them Up to Eleven then asks the question "how can we make this more unstoppable?" And manages to do it too, taking on multiple ships each five to fifteen times her size and winning.
Ben Sisko. In the episode "Paradise", he's tortured by the Evil Luddite village leader by locking him in a metal box in the hot sun. When he's let out, she offers him water if he'll change out of his uniform. He responds by staggering back over, barely able to walk, and climbing back into the box. It's small wonder the Defiant is as tough as it is, given that he designed it in the best possible mood to design new warships - utterly furious.
It shouldn't be too terribly shocking, considering what she is, but Cameron of Terminator: The Sarah Connor Chronicles is pretty darn tough. Little things like getting caught in a massive car bombing don't do much more than give her a limp for a bit.
Cromartie. Motherfuck, Cromartie. He didn't even stop when his head was severed from its chassis and sent into the future. What did he do? Why, he sent out a remote call to the body that had been resting in a junkyard for eight years, put himself back together, and chemically reproduced new skin. Yeah. They had to destroy his processor and bury him before he stopped coming back.
There is also Allison Young, the girl whom Cameron's appearance was based on. She doesn't surrender, repeatedly attempts to escape, and when Cameron is about to kill her, she stares her right in the eye and declares she'll never help them.
Then there's a T-888 that was sent back too early and accidentally killed a guy who was supposed to construct a building where his actual target is eventually supposed to give a speech. What does the T do? Adopt the dead guy's identity, found the construction firm and design the very building. The robot then immurs itself in a wall and goes into standby to wait many decades for his target to appear and give a speech.
Comedy example: Jack Donaghy of 30 Rock. Among other things, he's grown an inch and a half and defeated a killer peanut allergy on pure willpower.
Done in a very silly manner in Monty Python's Flying Circus's 'Upper Class Twit of the Year Show'. The commentator says of one particular Twit 'He doesn't know when he's beaten this lad. He doesn't know when he's winning either. He doesn't seem to have any sensory apparatus whatsoever'
The Closer's titular character, Brenda Leigh Johnson.
Borders on a deconstruction when her obsessive need to close a case involves putting it before all else—her boyfriend's investigations, the rest of the LAPD, her personal relationships, her life . . . she eventually learns that complete dedication to her case at the expense of all else isn't healthy. Repeatedly.
Amusingly, in the final episode of Kaizoku Sentai Gokaiger, the Red Ranger (himself one hell of a Determinator) says that he admires Earthlings because the entire race has this as their trait.
Charlie Crews from Life. As a cop that's been sent to Pelican Bay - a federal maximum security prison - which houses Level IV criminals and some of the world's most notorious prison gangs, he pretty much got targeted for all kinds of hell and abuse imaginable. For the first few years of his incarceration, he spent more time in the prison infirmary than in gen pop because they kept breaking every bone in his body. One time, they cut him up so bad he had to get 241 stitches. That's not even taking into account the beatings he received from the COs. Aside from the physical and mental abuse that's already been heaped on him, he also got sent to the SHU for a few years and that knocked quite a few screws loose from his head. He then decides that he's had enough so when he gets back to gen pop, well... Let's just say the inmates get introduced to a whole new Charlie Crews. He's finally let out from prison when, after 12 years, he gets exonerated. He proceeds to win a $50 million lawsuit against the city, gets back on the job as a detective despite the universal suspicion he receives from the LAPD, tracks down the murderer he served the sentence for, and (after deciding not kill the guy) brings him in and puts him behind bars.
Crews, in the second season, gets shot in the chest. After he recovers, he lies to the LAPD that he doesn't remember who shot him, gets the bullet that hit him and keeps it, melts it down later to make a new bullet, barges into the shooter's house and proceeds to shoot him w/ a gun loaded with that bullet. It. Was. Awesome.
The time when his partner got kidnapped by the psychotic Russian mobster known as Roman Nevikov? Yeah, that whole episode was epic. Despite the entire LAPD looking for him as his connection to Roman became known and his only assault weapon getting confiscated, Crews meets up with Roman and exchanges himself for his partner. He gets into the SUV completely weaponless, Roman sitting beside him and 4 or 5 heavily armed Russians surrounding them. It was not Roman's wisest move.
Charlie: "Do you know how I survived 12 years in prison?"
Roman (sneers and laughs): "Your Zen?"
Roman's laughter gets interrupted when Charlie reaches over and punches him in the throat.
Charlie (watching serenely as Roman chokes to death): "Like that."
"Avatar" plays on this when Teal'c is trapped in a virtual reality game that's programmed to shut down when he either beats the scenario or uses the emergency exit. Except subconsciously, he never gives up, and so the game forces him to play it to the bitter end.
All of SG-1 qualifies, really. Throughout the entire series, there doesn't seem to be a single moment when they didn't. Their Determinator status is so legit.
For a villain example, you can't do better than Apophis. How many times has that guy died? And yet he still wants to kick Earth's ass to the bitter end.
Anubis fits the trope as well. After his shell is destroyed, he sticks around in Earth's orbit until he finds an unfortunate Russian cosmonaut to possess. Later, another Russian commits a Heroic Sacrifice by willingly stepping through the stargate to become a Human Popsicle. Somehow, this half-ascended bastard still won't quit. The only way to stop him is to force his former teacher to engage him in eternal combat.
Also works for Ba'al and Adria, with the former creating dozens of clones of himself to trick his enemies.
Hell, *everyone* on Destiny, just to survive another day. That's kind of the theme of the whole series!
Sherlock: Sherlock, despite his kookiness, has solved a lot of cases-and, hell, even if he's had to go to the GOSH DARN DRAWING BOARD he'll solve it. But he's even more determined to keep those close to him safe. No matter what.
John too. He's very determined to do anything for those he cares about. Also, due to his adrenaline junky-ness, he's willing to help pretty much anyone, really.
Blair Waldorf definitely qualifies. If she sets her sights on something, she'll stop at nothing.
Chuck Bass can be that way too. Which makes things complicated when he's determined to have Blair, and she's determined to have his best friend.
Even more complicated when they are both determined to get the other to admit that they're still deeply in love, while also determined not to let the other one know they still love the other person. Heck, just writing it down was kind of complicated.
On the Reality ShowSolitary, this is the entire gimmick. If you're the first person to quit a challenge, you lose, and since you have no idea how everyone else is doing, your only chance of winning is to just keep going.
John Crichton from Farscape. In fact, it's canon that his strongest trait is determination.
Noranti (to John's nephew): I like... that you're always striving to reach higher. Hoping for a better tomorrow. It's the quality that first attracted me to your uncle.
Bobby: That humans dream?
Noranti: Yes! You're so ignorant, but you never give up. Even in the face of insurmountable odds.
That could be one of the qualities that the Eidelons saw in humans and why they transplanted them to their own sector, artificialy "evolving" them into Sebaceans and charging them with enforcing the peace.
Claire Bennet from Heroes is a non-violent version. She's immortal, immune to pain, and effectively fearless.
Leslie, delirious with flu and barely able to walk, escapes from the hospital to give a masterful speech to the Pawnee Chamber of Commerce drumming up support for a community project. The awesomeness is in no way diminished by her immediately, upon the speech's conclusion, becoming completely incoherent again
The Cousins in Breaking Bad, out to avenge their cousin Tuco's death, cut a wide swath of murder and meyhem in their attempt to kill Walter that ends in a shootout that leaves Hank badly wounded, one cousin on the wrong side of the Chunky Salsa Rule, and the other with the lower half of his body crushed. When he later sees Walt in the hospital, surrounded by cops, he yanks the IV out of his arm, rolls out of bed and drags his bloody stumps across the room, never taking his Death Glare off of Walt.
Fox Mulder from The X-Files. As Scully put it, "...they could drop you in the middle of the desert and tell you the truth is out there, and you'd ask them for a shovel."
Law & Order: Trial by Jury gives us the best Determinator in the Verse, Tracey Kibre. She's Bebe Neuwirth. She's the Bureau Chief from hell. Distill all of the awesome of all the other ADAs into one woman; you get Tracey Kibre. Never. Gives. Up. Ever.
On season 6 of The Amazing Race there was Lena. While doing the infamous hay bale Roadblock, she unrolled hay bales for almost 10 hours, long after all the other teams had come and gone. She unrolled over one hundred bales without finding a clue, and did not stop until Phil came out to the field to eliminate her and her sister.
Due South presents us with Benton Fraser, who is such a Determinator that he makes others into Determinators even when they think they don't want to be.
Sue from The Middle. No matter how many times she fails to make any team or club, she never gives up. This extends far enough that, in that Season Finale, she "runs" five laps around a cross-country track with a twisted (and possibly broken) ankle, only for it to start raining while she's on her final lap. Then, after being splashed with mud and grass and losing a crutch, she drags herself across the finish line with only her arms.
Jack from Lost when it comes to saving fellow survivors. Notable examples include his attempts to resuscitate Charlie after he's hanged from a tree and pouring his own blood into Boone after his accident.
There are multiple examples of this in Hell's Kitchen. In season 2, there was Heather West, who seriously burned her hand early in the season. However, instead of giving up, she had a medic check on her hand, put it in ice water, then started to direct her team to help them cover her station, as she was unable to cook, and did not leave until forced to by the medic. To an even greater degree, season 6 had Dave Levey, who broke his wrist in the third episode. Not only did he keep going and cooking with one arm in a cast, he ended up winning that entire season literally single-handedly, telling Chef Ramsey in the final four, paraphrased, "Chef, I don't want you to take me out of this competition because of this wrist. If you want me out because I fuck up, that's fine, but don't take me out because of my injury." There is a reason those are two of the fans' most well-liked chefs in the run of the show.
The undaunted police inspector Corrado Cattani from the Italian TV series La Piovra (The Octopus) may be the walking embodiment of this trope. They have blackmailed him, threatened him, set him up, killed his friends, colleagues and even his daughter and his wife, but all that made him even more determined in his crusade against the The Mafia.
From Big Brother 6 (US), the houseguests were put in a competition called the "Pressure cooker". It was not physically taxing in the least bit, being a challenge where they only had to hold down a button and not touching the ground with anything but their feet. Instead; it was an endurance test of willpower moreso than physical strength. They all made it at least six hours...before the first person was eliminated. The houseguests were literally out there into the next morning before Kaysar just gave up.
Chang on Community, when it comes to joining the study group. One of his stunts involved him dancing for five straight hours so he could join.
Murdoc in MacGyver. Over the run of the series he is "killed" in practically every episode that he appears in, yet will always return to complete his ultimate goal of killing Mac. In the western themed dream episode "Serenity" from season five, he actually refuses to not complete a hit on MacGyver even after Pete has called it off, commenting that he has a reputation to uphold. Though these events are not "real", the attitude is clearly Mac's own recognition that he's being chased down by a rabid Determinator.
Gibbs from NCIS. He willget his man, no matter what. Shoot him? That'll heal. Hit him with a car? Ditto? Threaten his job? You can't, and he doesn't care anyway. Blow him up? Been there, done that. Twice. Legal protection? screw that. the list goes on and on and on and...
Penelope Garcia from Criminal Minds. Despite being totally scared, grossed out and freaked out over and over she refuses to quit and find another job. No firewall, password, IP hop will stop her from getting her data. Even when she is shot and almost killed by the bad guy she is determined to do her job.
Veronica Mars: A few characters potentially, but Veronica herself definitely fits the best. Let's list all the things that happen to her throughout the show: her boyfriend dumps her without notice, her best friend is murdered, she's rufied and raped, she's framed for cheating multiple times in an academic context (which is an efficient way to sabotage an academic career), she's locked in a burning fridge, attacked with her own tazer, and roughed up by an Irish mobster. That's not to count the relentless bullying an emotional isolation the rest of her school puts her through. And her response to all this? Find out who's responsible for [insert travesty] and make them pay.
Lieutenant (and later Colonel) Viktor Burakov of the Soviet Provincial Police, in Citizen X. The head of the FBI's Serial Crimes Task Force calls Burakov "the one man in the world [he] would not want after him, because he absolutely will not give up."
Tom and Ian made the final immunity chalenge (an endurance challenge where they held onto buoys) last almost twelve hours.
In a "Double-individual-immunity" challenge where a man and a woman would win immunity, Jane decided to continue holding onto her weight until the men were done.
The first individual immunity challenge in Australian Outback lasted about ten hours.
Eliot Spencer of Leverage, who at one point threw down with a vicious Psycho for Hire while already suffering from cracked ribs and a concussion. The fight ends with Eliot badly winded, bleeding from the mouth, even more severely injured...and victorious.
Eliot himself had to face one of these in "The Schaherezade Job" in the form of a Giant Mook whose flat out refusal to go down verged on Implacable Man. Eliot knocked him out and the mook got up and came after him again. Eliot fights him again, and this time, can't knock him out, so he and Parker blow the floor and drop the mook into the vault below, rendering him unconcious again. The mook wakes up a few minutes later and prepares to start the fight again before being waved off by his boss.
The eponymous Merlin. He's an Iron Woobie and Hurting Hero turned Up to Eleven who has been through hell and back barefoot, and he's still continuing on his mission to free the magical people with all the conviction he had from day one. If anything, having all the horrible stuff happen to him has made him even more determined.
The Big Bang Theory : Being a Spoiled Brat, when Sheldon wants something, he'll just never give up until he gets it, no matter how much he needs to annoy his friends. And most of the time, he does get what he wants.
From V (2009), there's Georgie. His ability to resist Cold-Blooded Torture is all based around his utter rage and hatred of the aliens, and his ability to not break comes as a surprise to the aliens' torture experts.
Friends: Monica Geller. Insanely loyal and stubborn, never get in the way when she's set her mind to something. Examples includes losing something like half her body weight in a year to get revenge on her future husband, enduring horrible bullying to prove herself at her job and winning any game she plays through sheer will-power. This probably stems from her difficult childhood, where she was overshadowed by her older brother and frequently put down by her parents. To get anywhere, she had to work for it. Just getting past her mother's criticisms and emotional abuse, belies her determination.
The Winchester brothers in Supernatural. Throughout the series, both brothers have been killed by angels, demons, demi-gods, monsters, and other people over and over and over again, yet keep managing to find ways to resurrect themselves, whether it be through demonic pacts, supplication to various angels, or by negotiating with Death himself. As long as one of the brothers still lives, he will (except for that one time) do whatever it takes to bring the other back to life.
Patricia in House of Anubis, at least in the first season. She would stop at nothing to find Joy, only taking one break when she believed that Joy was safe, which lasted only an episode. In fact, almost everything she did was to find her friend, and she did a lot in that season.
In fact, Sibuna as a whole counts. No matter what they have to go through- and there's a lot- they keep on pushing through, knowing that if they don't finish their quests, there will be grave consequences.
Jerome was this, when searching for his father's gem. He vowed to find the culprit, and he didn't stop until he did.
Joy may not have been the nicest person on the show, but she was quite determined to make Fabian like her.
You'd think that Corporal Max Klinger of M*A*S*H would give up trying to secure a Section 8 discharge after the failure of, oh, his first dozen or so attempts. You would be wrong.
JAG: In "Desert Son", we learn that General Williams made three trips in a damaged helicopter to rescue a special operations team, after having most of his leg blown off and the rest of his crew killed. That's why he got the Medal of Honor.
The Toyota Hilux. Thrashed around Bristol, drowned in the Severn Estuary, set on fire, smashed through a wooden shed, crushed under a caravan, battered with a wrecking ball, and even placed on top of an apartment block that was subsequently blown up. Charred and mutilated beyond recognition, each and every time it was still able to start up and drive away, with only a little assistance from a mechanic who was not allowed any spare parts or advanced tools. That did more for Toyota than any clever advert or marketing campaign ever could. No, really.
For one special episode, they had to each buy a second-hand four-wheel drive car on the Internet, and drive it from the centre of the Amazon rainforest to the coast of Chile. All three of them reached the coast, and Hammond's car only failed to finish because it took a tumble down a sand dune a few miles from the finish.
America Unearthed has host Scott Wolter starting investigations in the middle of his vacation and other non-work moments. (Placed here instead of Real Life because it's obviously scripted, hence the presence of the camera crew).
Gotham: James Gordon, has shown qualities of this on multiple occasions but one that stands out is where he gets beaten bloody and knocked unconscious but upon waking up his first act is to reach for his gun and head to the police station, where he had been informed of trouble, instead of seeking medical assistance for his injuries
Street Justice: Adam kept up his search for Grady for 20 years, even putting a good portion of his police-detective salary toward hiring private detectives to find any trace of his former friend.
This trait is very common for Forged In Fire contestants. Several of then keep powering on after severe setbacks and manage to turn in finished work, and one guy managed to work through severe pain caused by arthritis. Doing this also tends to impress the judges.