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.
In another of Monty Oum's creations, RWBY, 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.
Deconstructed by Yang in her backstory. Desperate to find her biological mother after her stepmother (and Ruby's mother) disappeared, the very little Yang took Ruby through a giant forest, bearing numerous injuries along the way, to find her mom. By time she'd reached her destination, Yang was heavily wounded and exhausted to the point of near unconsciousness, just in time for a pack of Grimm to try and kill her and Ruby, and the only reason Yang's stubbornness hadn't gotten her and Ruby killed was because of Qrow killing the Grimm thus saving the two. Yang shares this story with an exhausted, rundown Blake, who was dedicating every minute of her life to finding Torchwick at the expense of her health and sanity, hoping that she would stop Blake from going too far with this trope.
Blake carries an unconscious Yang to safety right after having been viciously beaten by her abusive ex-mentor and stabbed in the abdomen.
Ruby Rose proves to be this by the end of Volume 3. two of her friends are dead, her team is broken up, her school's a lost cause, her sister's maimed and the people who did this are still out there. Yang's given up on everything, yet Ruby's been told where those responsible were last told and she heads off to stop them, flanked by the teammates of one of those fallen friends. Best emphasized by this quote during her final battle with Roman Torchwick.
Ruby: I don't care what you say! We will stop them and I will stop you! BET ON THAT!!
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 donít have to eat. We donít have to sleep. We heal ten times faster than ordinary humans. Weíre the United States military. Letís 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.
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.
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 Pokťmon 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 Trap music 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: Iíll keep driving this truck. Iíll keep wandering this country. Iím going to find you. I will.