Lucretia from The Adventure Zone is the anti-hero variety of this trope. She worked unrelentingly for a decade to almost single-handedly reverse the damage of the Grand Relics, protect her friends, and keep the Hunger away, although her methods were certainly questionable. This attitude and behavior are explained in the Stolen Century arc as being the result of a year she spent on the run from the Judges while they rest of her crew was imprisoned.
Tifa Lockheart in the fan film series Dead Fantasy is an insanely awesome representation of this trope. Throughout five shorts, she has been slapped with an axe capable of downing a skyscraper, beaten to a bleeding pulp, which involved literally having the Materia knocked out of her, jumped onto a train, been slashed quite a few times, filled with arrows like it was target practice, stabbed through the arm, and chained up by kusarigama. It takes Hayate's intervention to finally stop her.
Ruby is determined to make the world a better place for people to live in, no matter what kind of threat she's facing. During the Battle of Beacon, she is heavily outmatched by both Neo and Roman, but she doesn't falter. Even when Roman is beating her with his cane and ranting about her inability to accept the world is a dangerous place where heroes like her achieve nothing but an early death, she keeps standing back up. The world might be a cold, dark place, but Ruby refuses to give up; even when her team mates are forced by circumstance to quit the fight, she simply joins forces with Team JNPR and keeps on going.
Deconstructed by Blake. When Blake finds out that Torchwick is working with the White Fang, she becomes obsessed with finding out they're up to and stopping it. Her determination becomes an obsession that takes a heavy toll on her grades, social life and health. Yang takes her to one side to tell her a story about how Yang's obsession with finding her birth mother almost got herself and Ruby killed at a young age. Yang has since mastered her desire and helps Blake to do the same. Blake eventually plays the trope straight when she saves Yang's life by picking up Yang's unconscious body and running despite having just suffered a vengeful beating and being stabbed in the abdomen. Something that eventually keeps going in future Volumes.
Yang has been through A LOT to the point where she is one of the biggest Determinators in the Franchise. There is little she hasnt been put through. Abandonment, near death, public hummiliation in front of the World, actual dismemberment and even more abandonment by almost everyone she remotely cared about in her life. While she ends up nearly crossing the Despair Event Horizon after Volume 3 (having suffered the most of the surviving Characters from the Events), she still fights on, in spite of PTSD sets out into the World again to fight and protect her sister. Once this Girl is in the game she wont quit, not even when she has to Protect her Partner from The man who took her arm even though she is struggling with a Panic attack while doing so. No wonder Blake sees her as the embodiment of Strenght.
This is Jaune Arc's greatest strength. He lacks any combat background, unlike the main cast, which makes him one of the weakest characters in the series. Even so, despite the visible gap of ability between Jaune and almost every opponent he encounters, he has never backed down from a fight and, once he engages, there's no way to intimidate him or make him give up. Pyrrha once comments that she's never met anyone with as much drive to better themselves as Jaune.
Ozpin has been fighting Salem for thousands of years with seemingly no progress, but outright refuses to give up, putting his hope in humanity again and again no matter how often they fail; even his most loyal supporter, Qrow, criticizes his perpetual optimism. As he lives under a divine curse that prevents him truly dying until Salem is defeated, Salem cannot simply get rid of him by killing him. Instead, she states her plan is to destroy Ozpin's spirit through destroying all his hope for humanity's future. Ozpin has been tasked by the gods to unite humanity and prepare them for Judgement Day in an attempt to prove that humanity is capable of being redeemed. By dividing humanity and sowing chaos, Salem is making it impossible for Ozpin to fulfil his mission and, because she has been cursed with Complete Immortality, Ozpin cannot solve the problem by killing her. However, if the gods decide humanity is irredeemable, they will destroy the entire world. Humanity's survival therefore rests entirely on Ozpin's determination to keep going whatever the odds.
LordKaT is the brain behind the Until We Win videos, where he plays obscenely difficult video games and does not stop until he does in fact beat it.
By 2011, CardGamesFTW' - the YouTube account of LittleKuriboh - was up to five bans.
In a Meta sense, see the page picture up there? It went through the third biggest Image Pickin Forum with the crowner with the most number of images EVER. It was open for months (IIRC) without any examples with above zero points. But guess what, we made it, and there's the picture. The people who participated in the forum are all Determinators.
The New Avengers as a whole, and several other characters as well, in Marvels RPG.
The Nostalgia Chick lives in either LA or New York City. She will run from either of these places to Virginia and Chicago to stalk and fight (and chloroform), respectively.
On that note, The Nostalgia Critic had run from Chicago to Pennsylvania for a brawl with the Nerd. More personally, the Scooby-Doo review had Rob reveal he's been trying for five years to get Critic to stop wallowing and play a poker game with him.
In The Salvation War, the First-Death humans (those who have died and are now in hell) that become main characters are basically this. Extra points because they mostly are members of the US Army. As one of its characters puts it:
Liutenant Jade "Broomstick" Kim: Okay, guys. We dont have to eat. We dont have to sleep. We heal ten times faster than ordinary humans. Were the United States military. Lets go blow up some baldricks.
Fantendo's user Cloverfield Monster. Despite being lynched and told to leave, he never gives up on making games.
Zerosaga1 is known for having this attribute, even when playing extremely difficult games.
Church of Red vs. Blue is basically this. He acts like he doesn't care and isn't very good at being The Hero, but if there's something he thinks has to be done, he never gives up until he accomplishes it. This goes as far as to him facing Wash and the Meta at the same time, after being shot, with just a pistol and Doc to help, after Tex is trapped in the memory unit in season 8. It doesn't work, but he still tried.
"A Simple Walk (Into Mordor)", a set of videos featuring Chris and Kerry of Rooster Teeth setting out to walk the distance between (the shooting locations of) Hobbiton to Mordor. Despite the conditions they walked across, the blisters they busted, how incredibly poor their hygiene was and how many laws they had to break to get there, they did it in six days and buried the One Ring in the site of Mordor.
In Savage Divinity Rain has gained the title of "The Undying Savage" partly because he refuses to die and give up in combat. He also train's to ridiculous lengths and expects his retinue to do the same.
I give up on giving up. I am a gamer. I CANNOT walk away from this stupid ass boss. BUG OR NO BUG. I will win.
He then works out how to do it by changing resolutions during the fight.
Out of the many badasses inhabiting AJCO, the Shrinking Violet with a stammer is the most determined. Egg has been kicked down and beaten up - physically, mentally and emotionally - more times than is possible to count but just keeps going. No matter how many times her attempts to help have gone sour, she will not stop trying to help. She gets just that little bit stronger every time she picks herself up - she'll probably be a Badass Pacifist when she's older.
The Mob of Twitch Plays Pokémon do not stop until they have defeated the Champion. No matter how many Pokemon are released, no matter how many times they lose, and no matter how powerful the enemies in front of them become, they just won't stop. As a side-effect, their hosts come across as this in-universe - Red spent days in Team Rocket HQ, AJ spent three and a half hours to get out of the first route in the game, and Napoleon spent eight hours in the Great Marsh trying to avoid disaster. Camilla A Slash gets the prize, however, for fighting the Elite Four over one hundred times.
The girlfriend of the Kaizo Trapmusic video, after 61 on-screen deaths (and many more off-screen, judging by the bloodstains) hits a game over on the penultimateboss - the boyfriend she's been seeking the whole time, who killed her because she dropped her guard after seeing his silhouette. On the Continue screen, she's given the option to either return to the beginning of the very first level, or leave the game and return to the safety of the real world - without her prize, and with no guarantee that she can return. The game she's trapped in is a Platform Hell, and she has no tools or cheats she can use to make the playthrough easier. She weeps as she looks between the two options for several seconds. And then she chooses Yes. And she's been trapped in that video game for months. At least, if the state of the house in the ending is anything to go by.
Movie Rehab: Sag does this show since 2013 with only few episodes and even less viewers, yet it doesn't stop him from working on his reviews.
Sag: I may be a nobody amongst thousands of internet reviewers, but that won't stop me from doing what I love to do.
The long haul trucker Narrator of Alice Isn't Dead is stubbornly persistent in her quest to find her wife Alice, to get an in-person explanation for the mysterious, Conspiracy-laden circumstances around Alice's disappearance, even though this search draws her into unwitting, unwilling encounters with the paranormal.
Narrator: Ill keep driving this truck. Ill keep wandering this country. Im going to find you. I will.
Levelengine plays Super Mario World ROM Hacks savestateless, only using savestates on Kaizo hacks. Even then, Game Genie is never used unless absolutely necessary.
Roadkill's hosts, David Freiburger and Mike Finnegan, are this to an absurd degree. Design flaw in the crazy car build that they're attempting to put together holding up the project? Use hackery and jury rigging to make it work. Project car breaks down? Do whatever it takes to get it back on the road. Show up late to the event that they're participating in, or miss it altogether? Do gratuitous burnouts and donuts in the project car anyway. Do it all again the next episode. You can't keep these two down.
DSBT InsaniT: Seth. In his own words; "Its gonna take more than an ass-kicking to kick my ass!"
Keela can be sent to an almost literal Hell, and she'll still stay strong.
Deviant: Several characters are this, including the main character, who literally has a power that gets stronger the more willpower she's running on. Tsunami is notorious for this, too, something aided by her incredible endurance.
In some Pokémon playthroughs, Pikasprey'll set some utterly brutal challenges for himself, but won't stop trying until he either succeeds or can no longer continue.
Worm has several, including the protagonist, but a few stand out even among these:
Chevalier, who at one point takes a laser to to gut, wakes up an hour later having had major battlefield surgery to stitch him back up, and promptly grabs his sword and armor before heading out to challenge Behemoth, one of the deadliest entities in existence, to a swordfight.
Taylor, who in spite of being cut in half first controls her landing so as to merely break her remaining limbs hitting the water, then invoke an Emergency Transformation device to return to the fight, and then continue fighting even as (late in the battle) the transformation is wearing off and her core organs are falling out of her abdominal cavity.