Ga'ri in the early parts of Big Sister, still trying to recover from the Gadget Drone incident mentioned in StrikerS carries a baby across a volcanic Death World with what little power she has left, crippling herself again in the process, then manages to fight off a lethal infection and regain use of both her legs and her magic, this time without the help of high-Magitek rehabilitation facilities.
Shinji from Children of an Elder God is pretty pliable and passive, but he can become a determinator when something important is at stake. In chapter 8, he wouldn't stop fighting in spite of his enemy's overwhelming power because he would not fail Asuka and Rei.
Moriarty is unbelievably determined to survive at any cost...
Moran spends several months hunting a seventeen-year-old girl. Considering that he spent over two years hunting Holmes, however, the surprise is not that he kept up the hunt but that the hunt kept him up.
I would fight to to make that bright and innocent past our future again. Even if it means dashing myself against the evil and cruelty of this wasteland until there is nothing left of me. And then, when I am too broken to go on, I will float my dying body right down into the throat of the darkness and make it choke on me.
Or, you know, this could all end in sunshine and rainbows. No need to get pessimistic.
Hybrid Theory. Determination is sufficiently central to the magical system and core plot that little can be said without spoiling the whole thing, except for "Holy crap!" and "Did you just punch out <censored>?"
Being Superman, Shinji does not know the meaning of the word “quitting”. A giant alien monster whose body is an alternate dimension has swallowed his Love Interest and is MindRaping her? He’ll dive into Leliel’s body, enter the illusion, convince Asuka to let him into her mind and burn the alien. Another giant monster with devastating power is flattening the Geofront and the army is about to blow it up with nukes, killing everybody? He’ll pick Zeruel, lift it and will take the impact of dozens of nukes. An ancient God of Evil has set Third Impact off? Shinji’ll wrestle Instrumentality’s control off Darkseid’s hands and will use that power to punch him.
Rei does not care about being badly hurt. If she has to fight, she will do:
Hearing the commander's voice, she stirred. Her one good eye opened and scanned the room fitfully, unable to focus. She settled her blurry gaze on the fluorescent lights above her head and waited as long as she could for the throbbing in her skull to ebb just a bit. It didn't, but she had to answer. She was driven. It was her purpose, her reason for being. Pain was irrelevant, a signal that her body was damaged, nothing more.
Superwomen of Eva 2: Lone Heir of Krypton: Asuka hates giving up because in her mind it means she is worthless and useless. It did not matter how hard the fight is or Whether she is fighting as a mecha pilot or as Supergirl. She grits her teeth and fights on. Sometimes she was tempted to stop being a super-heroine because she thought herself unworthy, but she always returned to fight when she was needed.
In Child of the Storm, this could apply to all the heroes — who never give up, no matter what — but special mention needs to go to Carol Danvers, as during the Final Battle it's revealed that she is literally too stubborn to die.
In Code Geass: Mao of the Deliverance, Mao definitely qualifies. He gambles, blackmails, steals, does drugs, flies through a frickin hurricane, takes on soldiers and terrorists and gets shot repeatedly all in his quest to save C.C. and be with her forever.
Takato , OH DEAR LORD TAKATO. If continuing to fight againgst a hoard of Mega level digimon. despite the fact that your clothes are literaly soaked in your own blood, and you are almost paralysed from your brain cells dying doesn't count as this, nothing does.
Daemon as well.
Takeru: "Daemon is an extremely powerful foe, and his power has increased in the past fifteen years. His anger and desire for vengeance makes him practically invincible, a berserk fighter who will do anything to win."
Takato: "But…my friends will never give up! Rika will never give up!"
In Dragon Age: The Crown of Thorns, the Wise Prince protagonist keeps on going and pulling one plan after another, even though the fact he grew up in a Deadly Decadent Court could have long ago turned him into a self-centered cynic. He also displayed a high physical resilience in the deep roads, up until he stumbled upon a certain someone else and finally decided to allow himself to faint after two days of no sleep, fighting, getting repeatedly injured, tried for fratricide that never happened, dragged through streets in chains, spat on and generally not having a very nice time.
In the Firefly fic Forward, Mal shows his canon tendencies, but on top of that there's Book, who pulls a You Shall Not Pass when defending River from Reavers. As the story progresses, River herself becomes one as well, up to the "Fourth Interlude" chapter, where little things like a bunch of cracked ribs or having a leg broken, getting shot in the stomach, and getting a concussion from a ricocheting bullet don't stop her from kicking ass.
Jericho has the eponymous narrator, Jericho, a stallion from the nation of Teutschland. He appears to be this, since he walked to Equestria, where the story begins, all the way from his homeland of Teutschland, which is across the world and several oceans from Equestria.
The Party Never Ended has the poor addle-sounding Derpy Hooves determined to fight the supernatural storm, no matter what. And the climax features a Battle in the Center of the Mind. In one corner, the entity that drove Luna into Nightmare Moon via Break the Cutie. In the other, Pinkie Pie with Twilight Sparkle helping. Then Rainbow Dash with Pinkie Pie helping. Then Pinkie Pie by herself. Break The Breaker Of Cuties, anypony?
In the Pony POV Series, the Mane Cast and their Sixth Ranger Trixie stop at nothing to defeat Nightmare Whisper, Fluttershy's Superpowered Evil Side, and save her from herself. By the end of the battle, Pinkie Pie and Applejack are severely cut up by glass (Pinkie Pie from being throw out of a glass window and Applejack from having glass daggers embedded in her back), Twilight and Rainbow Dash have both been sent dangerously close to the Despair Event Horizon, and Trixie has been electrocuted and broken one or more ribs. But the one who takes the cake is Rarity, who has her legs broken and keeps herself in the fight by moving her legs with telekinesis! And thenFluttercruel fights her way out of a Jury of the Damned, getting pretty much mauled by demonic creatures, to get back to help save Fluttershy. She then proceeds to tear the tubes connecting her to Nightmare Whisper out 'by hand, despite the agonizing pain it causes.
This is actually a distinct plot point from a "side story" (Alternate Universe, but declared part of the POV multiverse by Word of God). An alternate Discord notes that the Ponies of the "Reharmonized" timeline, "Would never give up, no matter how hopeless the situation."
Also happens with Queen Cadence in the Dark World Timeline. She promised Twilight Tragedy she'd redeem her, but was vaporized before she could. Five hundred years later, her spirit appears to Twilight when she's suffering a Villainous B.S.O.D. and helps guide her back to the light, just like she promised.
Dark World!Derpy. To put this in perspective, she endured 1000 years of living Hell so Discord would remove the curse from Dinky. She marched around Discord's palace, assaulted constantly by crows trying to eat Dinky, and according to Dinky herself, she never let them touch her. And she did this with crippled wings and legs. Is it any wonder at all she becomes an Element of Loyalty? After she's healed by the redeemed six and Dinky is restored, this determination extends to protecting her new friends. Prime Derpy is equally determined. In fact, Derpy's cutie mark is implied to represent her undying will, and alternate timelines were she became a deity normally describe her as Goddess of Determination (and Mailmares), meaning Derpy is the ULTIMATE example of this trope in the Pony POV Series.
Sunset Shimmer, from Seven Days In Sunny June, is willing to face down date rape gangs, nightmarish unicorns and a death threat from Princess Celestia if it means saving those she loves.
In Just Before The Dawn, this is one of Tercio's defining characteristics, and what brings him to Celestia's attention. No magic, super strength, or flight, just determination and willpower.
In The Witch of the Everfree, when Applejack is on a quest to kill the timberwolf that killed her parents, Sunset Shimmer gets this line:
She gave me a determined look, the kind only a foal with no sense of self-preservation could have. I was pretty sure I’d given ponies that exact look a few times.
In Stroll, Octavia will not give up on trying to get home. Best shown by this line in the second chapter:
She would. No matter if it took hours. Or days. Or weeks. Or even months. She'd find her way back home.
Naruto in The Sealed Kunai. While fighting Hidan and Kakuzu, he holds Kakuzu inside his barrier trap jutsu for one full minute while being sympathetically mutilated by Hidan, who is specifically trying to break his handseal. It eventually lapses when Hidan actually stabs through his hand, but enough time was bought for help to arrive.
In Vapors, in addition to the canonical examples, Aiko is repeatedly held up as an example. Due to her high intelligence and maturity, her apprenticeship instead of being on a genin team, and her past successes, Aiko has been unintentionally conditioned to soldier on through horrible circumstances in pursuit of her goal without asking for help. During her time infiltrating Root she nearly has several breakdowns due to the workload and emotional toll, but continues developing her plan alone for months. Then a new development means that she needs to end things right now, at which point she recruits Sai to help and finishes things in less than a day.
Before the beginning the story Shinji had stopped hoping for him and Asuka getting together, but he never stopped wanting. His perseverance finally paid off.
After being told by his wife that she will not forgive him unless their children and Asuka forgive him first, Gendo is determined to earn their forgiveness, not matter what. Even if it seems all but imposible he is determined to do whatever that it takes, and he is willing endure anything.
Advice and Trust: Rei. Asuka compares her with a Terminator because she never stops. Her finest showing in the battle against Zeruel is actually an example of her Character Development, where she fights her hardest to defend her friends rather than lack of regard for her own life, never gives up and thanks to her resolute attitude she manages killing it in spite of horrible odds.
When Asuka ran away, Shinji finally gave up on her? No way! He searched and found a lead, drove one whole day –even though he had no reasons to think that she would remain in one place for so long-, and despite of her stubborn refusals to return, he proposed her.
Ghosts of Evangelion: Shinji was determined to take care of Asuka and never run away again after Third Impact. He kept his promise, no matter what.
HERZ: Asuka never stopped fighting, not even when the MP-Evas impaled her robot and she felt the pain of getting chopped into pieces and eaten alive. During that ordeal her heart stopped beating... and then it restarted. She survived, recovered and even managed delivering a healthy child.
In Neon Genesis Evangelion: Genocide, Shinji becomes determined to stay with Asuka and never leave her alone, not matter how much or how harshly she tries to push him away.
In Once More with Feeling, Shinji is sick of running away after going back in time. Now he’s got a second chance, he’s determined to never giving up or quit.
The One I Love Is: At the end of episode 11, when Shinji is at his lowest, Misato explains why he can not and must not give up. Thanks to her words he does not give up and he keeps fighting during the Final Battle, even though everything seems lost and he has nothing left, and defeats nine war mechas. And because he did not give up, Asuka survived the battle.
Shinji became one after Third Impact in The Second Try. Since running, hiding and complaning had got him nowhere, he decided to make an effort to be more determined and resolute.
Thousand Shinji: One of the key components of Shinji's ideology in this story is keeping hoping and never -ever- giving up. He also instills that way of thinking into Asuka (who refuses to give up and stop fighting, hence she does not try to kill herself during her fall into despair as in canon) and Rei (who endeavours to endure, not matter what).
During the climax, Misato dies and he loses hope for first time. Then Asuka gets killed and he gets broken. When Rei comes along and tells she can bring them back, Shinji regains hope and he remember that he never, ever, quits. And if he has to storm the Afterlife to rescue Asuka and Misato, he will do so.
Ruby in the Gorn-laidenBlood Rose becomes one of these - of note is the end of chapter 22, where despite having lost both her legs and had most of the skin on her body burned off by an incendiary barrier, she starts crawling off to escape before her Healing Factor even begins to kick in.
Nabiki in the Dark FicIll Met by Starlight. No matter how terrified of Ranma (who here was turned in a psycho by the Nekoken) she is, she resists the impulse to run away just to try and save Akane by him, fully knowing it could end with her having an 'accident' or worse. In the end Ranma does push her under a car in retaliation for her trying and nearly succeeding at killing him, but not even that stops her: while she's in the hospital dying one of her associates delivers Akane her last message, one that makes her finally realize what Ranma really is, and Nabiki just proves so stubborn to survive in a coma for months when she should have not survived the night.
[Holmes] would have died for a secret. But he will live now, for a friend.
Watson: "Porlock, there is no use in arguing the point. I shall search for Holmes. Whether you help me or not is your choice, but you shan't stop me — you have my word on that."
Wiggins: "Mum, I've been running that risk ever since I was a mite. I can't tell you what I'm doing, but I need to see it through. Peter's a fine young man, and he'll take good care of you. I'll come back if I can, I promise."
Optimus and Starscream in Hate the Dark, but mostly Optimus.
Eugenesis has Galvatron. Imprisoned, damaged and implanted with technology designed to prevent him from transforming ever again, Galvatron still manages to break free, and more importantly, transform, utterly ignoring the fact that the pain and strain of doing so hurts more than anything he's ever experienced.
Quicken: Emma becomes this after dying and being reborn because giving up got her noseless and dead. When she realized she was buried alive, she broke her coffin and then she dug her way out of her grave out of sheer stubbornness and despair.