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:
Lieutenant 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.
"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.
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.
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.
Can You Spare a Quarter?: "Nothing short of wild animals" is going to stop Graham from going back to Jamie after their second encounter.