Every video game protagonist or antagonist will wind up being a determinator, if you really think about it. In-universe, no-one sees the thousands of deaths, extra lives and continues; just Mario relentlessly, mercilessly crushing everything in his path.
Depending on the difficulty of the game, the players themselves will end up as determinators if they beat it.
Gordon Freeman of the Half-Life series. He's teleported to another dimension as the test chamber he's in is destroyed, then he climbs all the way to the surface past hordes of monsters to get help, finds out the government is cleaning up the mess by killing everyone involved, and he keeps moving forward without a clear purpose in mind until he's told about the Lambda team and eventually goes into another dimension to kill a gigantic telepathic creature called the Nihilanth. Then in the first sequel, he finds out that many years have passed since that and he goes up against a group of oppressors who have conquered countless worlds and dimensions and have already pretty much conquered Earth after a brief 7 hour war — yeah, just 7 hours to conquer Earth — and he gets a messianic reputation from both the events at Black Mesa and the major bulldozing he gives the Combine. Doctor Breen has to give little speeches to Combine soldiers where he berates them and points out that Gordon Freeman is just one man, but it doesn't do any good.
In the more traditional sense, Super Punch-Out!! has Gabby Jay! (YAY!) His all-time fight record is one win and ninety-nine losses, he's been demoted to a jobber judging by his being the very first fight of the Minor Circuit, and he happens to be one of the oldest fighters in the entire game at 56 years old. Despite all of this, he insists that he's got it in him to win just one more fight, and won't even think about retiring until he does.
No More Heroes 2 has this as a game mechanic as well: After your HP is reduced to zero, you get a window of opportunity to furiously shake the wiimote until Travis recovers and gets back a couple units of health. This can even be done a few times per section (which you'll definitely need against certain bosses, and even moreso on Bitter Mode.)
There is a reason for which Jasper Batt Jr.'s second form is That One Boss: his instant killing attack triggers a cutscene of Travis' death, removing any chance of getting some last second wind.
Kratos of the God of War series. He's literally climbed out of Hell... four times. As he puts it: "If all of Olympus would deny me my vengeance, then all of Olympus must die!" The fact that his only clear motivation is the desire to kick the ass of anyone who screws with him just solidifies his status as an undiluted Bad Ass.
By the end of the third game, there is only one thing that has been sucessful at killing him: Himself.
Even then An after-credits scene reveals him gone from the scene of his death with a bloody trail showing that he crawled away.
Betrayal at Krondor gives us Gorath, a moredhel (dark elf) chieftain who seeks to save his race from further destruction in another war they would start and ultimately shape them into "more than savages". And he flinches at nothing in his quest to do so. Surrender himself as a messenger to the humans his people have waged bloody war against for centuries, with his own kind branding him traitor and no reason to expect anything better than torture and execution at the hands of those he seeks help from? No problem. Go to Elvandar, the home of the eledhel, where a moredhel could expect to be killed on sight, or, as one of the Returning ones, cease to be moredhel and instead be enlisted into the eledhel ranks, when even travelling there causes him physical agony? Sure. Let himself be teleported blindly to the location of the most powerful magician on two worlds currently in need of a rescue? Go ahead. Touch a powerful relic of the Valheru - the force he fears most for the madness it has caused his people - in an effort to stop the Valheru spirits from escaping, ultimately at the cost of his own life? A Determinator to the end.
The space marine from Doom utterly refuses to go down without a fight, despite the fact that all hell has literally broken loose, and he is the only non zombified person in millions of miles. He proceeds to blow up all the demons and zombified people that are attacking him. He then fights his way into Hell, kills everything there, and fights his way back out. And that's just the first game. In Doom 2, he indirectly destroys Hell. Talk about a Badass Normal.
He gets even more badass. In the Nintendo 64 exclusive Doom 64 (the last game in the Doom franchise before it was rebooted with Doom 3) one powerful "mother" demon resurrects everything Doomguy had killed in his previous fights. His response? Eradicate anything between him and the Mother, kill her, and then stay in Hell to make sure the demons don't try anything like that ever again.
Axel Almer in Super Robot Wars Original Generation 2 is absolutely obsessed with defeating Kyosuke Nanbu. The part that makes him a Determinator is that his actual beef is with the Kyosuke Nanbu from Axel's original dimension (he is part of an invasion from another dimension), this Kyosuke is known as "Beowulf" and repeatedly humiliated Axel in battle. The fact that this dimension's Kyosuke has no idea who Axel is or why he is so determined to defeat him doesn't seem to matter to Axel.
Original Generations mellows Axel's Determinator personality a little, but he's still intent on stopping Kyosuke. Though he just uses another reason. And when he's Back from the Dead in OG Gaiden and had a Heel-Face Turn, he seems to completely lose his Determinator status... until the Endless Frontier EXCEED where it shows once again: Despite Alfimi telling him to just leave her of getting sucked to a portal along with some slime-like giant monster, and the Soulgain is damaged, Axel will have none of that since he already made up his mind about repaying how Alfimi got him revived. He picks up a blade tonfa and make a jump to save Alfimi (on foot, not with his mecha), assisted with Kouta/Fighter Roar.
Also shown in OG Gaiden is probably his method of saving Lamia. Everyone else, even Kyosuke, just seems to be ready to just give up and accept in despair that Lamia will die. Axel? Tells everyone to shut up and just use his dangerous gamble of using Code DTD as a method to restore her, which by the way, has the side effect of any minor misstep will cost her memory. The result is that she is saved. When Axel got into the same situation with sister unit Aschen, not to mention he was being amnesiac at that time, he's still insisting on using it anyway and succeeds.
And speaking of Kyosuke Nanbu, whenever something happens to his partner and lover Excellen Browning, he himself becomes a Determinator, plowing through anything in his path to get her back.
And, of course, Sanger Zonvolt, despite being easily one of the most competent of the heroes and in a powerful machine to boot often finds himself outmatched, outgunned, and even out and out disabled. His response? "Shut up! And listen! I am Sanger... Sanger Zonvolt! The Sword that Cleaves Evil!"
Luca Blight, the main villain of Suikoden II is the embodiment of this trope. Despite being the prince (and later king) of a massive country, he is the single most powerful human warrior on his side of the field (and arguably, the entire Suikoden universe), turning the tides of entire wide scale battles simply by appearing and punishing/killing any of his men for so much as hesitating in battle. If more proof is needed, his death scene should make this trope obvious. This trait, combined with the fact that he is a sadistic Omnicidal Maniac and a Nietzsche Wannabe, makes him a very intimidating and frightening villain.
Deconstructed in Tales of Symphonia: Part of what made the Knight Templar main villain what he is was that he could not, in any way, manage to give up on his ideals as they became more and more warped and admit that there might be a better way to do it - even as he lies dying, he is unrepentant and claims he would do the same things over again if given the chance to redo his life. This puts the villain in sharp contrast with Lloyd, whose ideals also clash with the way the world works - Lloyd, however, knows to yield and learn from his mistakes when it is obvious that he has done wrong, which is an integral part of the Character Development that turns him into a All-Loving Hero.
The "Unsent" from Final Fantasy X are people who've died but still cling to existence through sheer willpower.
Special mention must go to two Unsents Shuyin and Meachem who have done this for the last 1000 years before the party finally put them to rest.
Similarly the Undying from Final Fantasy XIII are Cei'th who were so enraged with the fal'Cie who made them l'Cie that they keep going forever out of sheer hatred and rage, rather then eventually fossilizing as most Cie'th do.
Final Fantasy VIII has Seifer, who refuses to back down no matter how many times he gets his ass handed to him, even after it's been made clear to him that he's being manipulated and used by the Big Bad.
To balance his determination, there's Squall, who develops into an immense Determinator in the third disc of the game. When faced with the only possibility of saving Rinoa after she goes comatose being crossing an ocean-spanning bridge on foot, he doesn't even hesitate, and carries her on his back, on foot, all the way across. Empty, monster-filled wasteland stretching on for tens of miles ahead of him? He just keeps walking. Finding himself inside a technologically advanced and potentially hostile country? Keeps moving forward. Finds out the only way to save her is to go into space? No hesitation to go up there. When she gets hurled out into the voids of space and all he's got is a quarter of an hour's worth of oxygen in his suit, and no real guarantee he's going to save her if he can get to her, Squall hurls himself right out after her. Nothing, and we mean nothing is going to stop him from saving her.
Final Fantasy VIII also has the X-ATM 092, better known as that giant frigging robot spider that chases you during the escape from Dollet. The damned thing pesters you the entire way, and the best most players can hope for is to either avoid fights altogether, or if cornered, fight it until they manage to flee the battle. Even avoiding the fights with it are a Guide Dang It, and guess what else? You're TIMED.
Squall's status as a Determinator is taken even higher in Dissidia: Final Fantasy, where his stubborness and determination are what keeps him going the most, forging his own path to end the conflict. He stubbornly refuses to work with anyone else, but also refuses to say why. Ultimecia believes it's because he can't trust people, but when he reveals that he trusts his friends completely, but would rather endanger himself to draw more enemies to him and save them the effort required, he quickly adds Bad Ass to the mix: he's trying to appear weak so he can destroy the enemies who think he's an easy target, so they won't go after his friends.
Terra, from Final Fantasy VI. With the defeat of Kefka, the source of magic is gone and all Espers are vanishing from the world. This includes the Half-Esper Terra. Even knowing this, Terra chooses to enter her Trance form and lead the party out of the collapsing Tower. In the final stages of the ending, after the last shard of Magicite has vanished, Terra is still flying in order to lead the airship out, even as her power wanes and she begins to fall. In the end, The Falcon catches Terra, who has reverted to a human form. Not only does she manage to outlive every other Esper on sheer determination, she manages to completely survive, albeit as a human.
Along the same lines, Locke has his girlfriend put in to suspended animation until he can revive her with something he didn't even know existed at the time. He spends most of the game looking for it, up to and including the end of the world. Of course, when he does find it his girlfriend decides to die and fix the Esper so that the party can use it. Deciding who you are and what your purpose is for yourself is kind of a recurring theme in that game.
Let's not forget Sabin. After the first fight with Ultros in a river, Sabin unsuccessfully tries to finish the old octopus off and gets flung off the raft. Washed away by the rapids, he winds up miles off course. His response? Start walking. He walks...straight through an Imperial military base, across a savanna teeming with monsters, across an entire ocean along an underwater trench, and to the afterlife and back. After The End of the World as We Know It, we find that he apparently never gave up hero work, limited only by the fact that he's just one guy. The first we see of him, he's holding up an entire house by himself, for five minutes. After saving the kid inside, his response is to immediately join your party, no questions asked, having never lost his enthusiasm for busting Kefka in the jaw. He also utters the line representative of the entire main cast:
Sabin: "Did you think a minor thing like the end of the world was gonna do me in?"
Cloud, from Final Fantasy VII, especially shown in his pre-game fight with Sephiroth. Sixteen years old, relatively untrained, no super powers or super strength, 5'7" tops (assuming he didn't grow between ages sixteen and twenty-one) and maybe 100 pounds soaking wet and fighting against The General - half a foot taller, probably at least twice his weight, genetically manipulated, trained from birth, insanely strong (and just insane). Cloud is knocked out while his town burns, but gives chase, eventually fighting Sephiroth, getting impaled by Sephiroth, and while still impaled on the man's blade and hoisted several feet in the air, somehow manages to get the upper hand and send Sephiroth flying into a pit of Mako. Not half bad!
Cloud being a Determinator is really his defining characteristic in the original game; when you look at his past he washed out of the SOLDIER program, indirectly cause Aerith's death, and spends a good portion of the second act of the game catatonic. Despite all of his failures, Cloud just keeps pushing through and refuses to give up.
Dying multiple times won't stop Sephiroth from trying to get even with him for that, so he counts, too.
The comparatively squishy lesser villain and corporate dictator Rufus might also count, given his unending pursuit of Sephiroth, only temporarily halted by explosions?
Zack Fair, in the Final Fantasy VII prequel Crisis Core, who escapes with Cloud after four years of horrible experimentation by the Mad Scientist Hojo. Zack steadfastly refuses to give up hope on or abandon the comatose Cloud, taking care of him for nearly a year, then has an epic Last Stand in which he battles a massive force from the Shinra army and manages to whittle them down to three troopers, in order to protect Cloud. And then, after being absolutely riddled with bullets, Zack still manages to give Cloud a Take Up My Sword speech before going out with a smile.
As for the Metal Gear series, just about everyone who has or once had "Snake" in their codename is one. Must run in the family.
For example, Liquid Snake is pretty determined to kill his brother. After having several Stinger missiles launched into his face, being inside the 30ft tall Metal Gear as it explodes, losing an intense fistfight on top of the 30ft tall Metal Gear, falling off said 30ft tall Metal Gear, getting shot in the face multiple times with a machine gun, and flipping over a Jeep going about 90mph (basically everything Solid Snake could've possibly tried in order to kill him), it's the nanomachine virus (that would've killed him instantly anyway, no matter the condition of his body at the time) that finally does him in.
Naked Snake in Metal Gear Solid 3 is a straighter example in that he continues the mission even when his old mentor (someone of which he described their relationship as being "deeper than love") utterly whoops his ass every time they meet.
Raiden in Metal Gear Solid 4 is equaled in his stubborn desire to get himself killed only by his inability to give up under any circumstances. A bisexual vampire incessantly stabs you in the torso? Just fight harder! A building falls on you and you can't move? Meh, you don't really need those limbs! Oh, now someone's about to ram a battleship into your best friend?Bring it!
And in Metal Gear Rising: Revengeance, after becoming an ever more powerful cyborg he finds himself weaponless and severely outmatched against the final boss, yet despite being literally beaten into the ground several times, he keeps getting up until a Big Damn Heroes moment from one of his allies grants him a better sword, evening the fight.
Raiden's predecessor, the Cyborg Ninja Grey Fox is almost the total embodiment of this trope. After all, he gets brought back from being killed by a land mine and takes on a walking assault tank armed with lasers, railguns, rockets etc. And even then only begins to slow down after having a third of his body sliced off.
Not to mention Solid Snake himself, who spends most of the time between later missions coughing up his lungs and trying to put himself back together with sticky tape. It doesn't stop him for a moment.
There's a point where he crawled through what's essentially a giant microwaved hallway, even if you lost all your health/psyche points, he will still crawl out... by his fingertips!
Ocelot - the only character to appear in all the MGS games - is stubborn enough to implant another personality into himself to carry on the plans his idol had for the world.
Volgin. Beat him up with your fists, grenades, shotgun, and whatever else you may have, and then blow up the weapons hangar with him in it? He blasts his way out with the Shagohod and chases after you, trampling everything and everyone who would get in his way, including flipping over a jet on the runway. Blow up the bridge he's crossing? He launches the Shagohod out of the wreckage and keeps coming. Pepper the Shagohod with RPGs until it finally breaks down? He punches his way out of the cockpit, uses his electrical powers to reanimate the blasted husk of a tank, and keeps on coming while you fire more RPGs and sniper bullets at his exposed body. And he STILL DOESN'T DIE until being struck by a lightning causes the bullets he wears all over his body to explode. Liquid would be proud.
The Boss. The woman was the strongest of a team that singlehandedly WON WORLD WAR TWO. She fought on Normandy, while PREGNANT! She had to have a C-section, and got shot during it! Then she got up and kept fighting. Even when she knows she's about to die, she's almost suicidal from the loss of The Sorrow, but still gives Naked Snake one hell of a fight in the end. That's a determinator if I ever saw one.
Captain Price. "The healthy human mind doesn't wake up in the morning thinking this is its last day on Earth. But I think that's a luxury. Not a curse. To know you're close to the end is a kind of freedom. Good time to take...inventory. Outgunned. Outnumbered. Out of our minds. On a suicide mission. But the sand and the rocks here, stained with thousands of years of warfare...They will remember us. For this. Because out of all our vast array of nightmares, this is the one we choose for ourselves. We go forward like a breath exhaled from the Earth. With vigor in our hearts and one goal in sight: We. Will. Kill him."
Soap qualifies as well. What do you do when you just went over a waterfall in an inflatable raft after holding it still so Captain Price could shoot the pilot of a helicopter? You drag yourself out of the water to check if Shepherd's dead. What do you do when you've been stabbed in the chest and Captain Price is apparently beating the tar out of Shepherd? Go crawl towards the gun Shepherd dropped. And finally, when you're laying on the ground, bleeding out from the knife wound and watching Shepherd beating Price into submission, what do you do? You pull the knife out of your chest and throw it at Shepherd, finally killing him.
Why don't we just say that all the main characters of Call of Duty series are an entire army of Determinators, ready to defend freedom and democracy against everything from Germans, Italians, French Nazis and Japanese to Islamic terrorists and Russian ultranationalists. Doyle, the playable British character from United Offensive and Call of Duty 3, was probably the Determinator of COD, until Soap took over.
Vhailor from Planescape: Torment, a Knight Templar so unstoppably dedicated to his cause that 'being dead' is a minor inconvenience to him, and also a really paltry excuse for trying to take a rest off from his duty of punishing the guilty.
Sonic the Hedgehog himself is a pretty fair example. Although it's often obscured by his eternal optimism and devil-may-care attitude, it's clear to see that he will never say die. The most notable example of his determination is Sonic and the Black Knight, where it's his will and determination that gives him the strength to hold his ground against the Dark Queen, even after she's beaten him to within an inch of his life.
Beautifully shown with one of Dark Bowser's attacks where he conjures shadowy versions of just about every type of creature in Bowser's army and sics them on the Koopa King. How do you combat this onslaught? By simply trudging forward, stopping every few seconds to block an attack or swat another minion out of your way, because nobody's going to stop you!
Also arguably Fawful. Lost in the first game, got blasted out the castle floating in mid air, apparently survived to run a shop in the sewers, conquered the world a few games later and got destroyed AGAIN to the point he's some kind of dark insect creature. THAT thing then gets destroyed in the final battle, and... it still tries to take Mario and Luigi out in a final attack. Summed up quite nicely in the ending:
Starlow: What? You're still at it? Look, give up already. Seriously, get over it. You're done with evil.
Fawful (partly): Fawful will disappear with no troubles. Forever disappearing... WITH YOU!
Emiya Shiro from Fate/Stay Night simply refuses to die (or stop) when someone he cares about is in danger, or someone is about to create danger for others, even when his opponent's power is several levels of magnitude higher than his. Slashed from shoulder to waist, overloaded nerves in half the body, broken arms and legs, cuts and slashes, loss of sight, flung into the air from wind pressure, thrown out third-floor windows, et cetera... he'll still keep trying.
All the above was just from the first route. The game has three, and the the third is far worse for Shirou. By the Normal End, he's a walking corpse powered only by his determination to destroy the Holy Grail.
Lancer actually has this as one of his skills; his ability Marshall increases his strength when he is in greater danger and makes it that so long as he has a body he will be able to fight. More to the point you can't Cherry Tap him to death because of his level with this skill. It would take a Noble Phantasm level attack to do him in. A shining example of this would be his death in the Unlimited Blade Works route.
Both the Master Chief and the Arbiter in Halo fit this pretty well. The Chief in particular literally does not believe in losing and will continue fighting against any odds.
Cortana: Just one question: what if you miss? Master Chief: I won't.
The Chief kind of has a thing where he won't die. The Arbiter's first and one of his last lines in Halo3lampshade just how difficult it is to kill John.
The Arbiter: Were it so easy.
There is a saying about the Silencers of the Crusader series: "Silencers get the job done." They might not survive, but they will get the job done.
Ryu Hayabusa of Ninja Gaiden. Within the first hour of the game, he is quite literally killed, but that doesn't stop him. Later, he is transformed into one of the demonic fiends he's been fighting against, and he doesn't miss a beat.
Gruntilda from Banjo-Kazooie. She gets pecked and shot with eggs several times, gets slammed by a high-powered jinjo around twenty times, falls off her tower making a hole in the ground, gets hit again when a huge rock slams down on the hole...and still struggles to get the boulder off. In the sequel, she's freed after having spent two years in the hole, having been reduced to a skeleton, but immediately blows up the protagonist's house. And that's just at the beginning of the second game. At the end of Banjo-Tooie, she is reduced to a skull. And proceeded to spend the next eight years rolling and hopping across the country solely to take revenge on Banjo and Kazooie. She's prepared to fight them even without any discernible means of doing so before L.O.G. turns up.
After Sparkster destroys most of its body, the final boss of Rocket Knight Adventures follows him off the exploding Pig Star and keeps fighting him as they re-enter Elhorn's atmosphere. It doesn't stop trying to kill him until it is vaporized by the heat.
In the Wing Commander series, the Kilrathi embody this trope. In fact, it's established in Wing Commander III that they literally don't know the meaning of the word "surrender" (even those few who are truly well-studied in Terran languages and culture seem to have trouble grasping the concept of it).
Skyward Sword takes this to extremes. From the few interactions Link has with other characters, it's apparent that saving Zelda is pretty much the only thing on his mind most of the time, and absolutely everything he does is working toward that goal.
Link is matched only by his greatest nemesis, Ganon. Turn Ganon to stone, lock him in a void, or just SLAY him, Ganon will rise again. It just doesn't stick. Of course, it helps that the Triforce bearers have some kind of mutual reanimation clause going on. In The Legend of Zelda: Ocarina of Time, when Link defeats him, he coughs up blood and stops breathing. Then, after Link and Zelda escape, he somehow comes back to life and turns into Ganon. After Link kills him for the second time by stabbing him through the head, he shows up again in the very next cutscene having apparently sustained no physical injuries and is just trapped in the Sacred Realm.
Demise, Ganon's precursor, has to be put on here too. Lost a humungous battle with a deity? Check. Sealed no less than 3 times by Link? Big deal- he'll just break free again. Got a freaking statue dropped on his head from above the clouds? Alright, he's dead- oh, no, wait: Ghirahim manages to kidnap Zelda and feed him her soul, allowing him to rise in his true form. Then, he is finally defeated by Link- takes the Master Sword straight into the forehead. What's left of him is sealed into the sword and permanently kept at bay by Fi. We all good? Nope- His final curse basically says that he is going to be reborn perpetually, forcing all of Link's/Zelda's descendants to deal with his reincarnations forever. He's basically the reason for every single Big Bad in the entire timeline- and also why it's so hard to permanently kill Ganon.
Zagi from Tales of Vesperia. After running into the protagonist Yuri in Zaphias Castle, he mistakes him for Flynn- whom he's trying to kill- and attacks him. After losing, Zagi continues trying to kill Yuri, even though he knows he's not the man he was looking for, because he was "the first man to ever make him bleed". You proceed to fight him 4 more times before the end of the game.
Notable catchphrase that he uses during his very first fight with Yuri: "I'll carve your name into my blood!"
Note that in his quest to kill Yuri, he goes down with a shinking ship, gets shot out of a giant fortress, trades his arm for a blastia arm, and drinks poison for another weapon against you. In one fight, you can knock him overboard the aforementioned ship mid-battle, and he will immediately, impossibly launch right back onto the ship to keep trying to murder you.
Probably the crowning example from the Tales series is Milla from Tales of Xillia. Prior to the game, she's never had to eat or even walk long distances - as the Lord of Spirits, her elemental minions did all the work. But as soon as she loses access to them, she learns to fight by herself, walks until she collapses from hunger, and plans to continue her mission even if she has to take on entire empires alone. Later, she's crippled to the point where she can't walk, but she's so determined that if Jude starts to dawdle too much she'll literally crawl away to finish her mission on her own. Damn, girl.
K.Rool, antagonist of the Donkey Kong series. Each boss fight with him requires him to take a massive amount of punishment before going down, and in the second game he survives being eaten by sharks.
Max Payne. Both games take place in a timespan of about 24 hours, during which Max hardly ever rests, is beaten up repeatedly, falls from serious heights, is shot and drugged on a regular basis and only really remains standing by virtue of dozens of dozens of painkillers. Of course, none of this stops him from going on a (mostly) solo Roaring Rampage of Revenge across town, until he's found and killed whoever mistakenly believed that killing a maverick cop with a deathwish was going to be easy.
Nicole Horne: What do you mean, "he's unstoppable"? You are superior to him in every way that counts. You are better trained, better equipped, and you outnumber him at least twenty-to-one. Do. Your. Job. Vladimir Lem: What the fuck is wrong with you, Max? Why don't you just die? You hate life, you're miserable all the time, afraid to enjoy yourself even a little! Face it, you might as well be dead already. Do yourself a favor, give up!
Count also Lieutenant Jim Bravura, Max's superior at the precinct, when he gets gunned down by a submachine gun at the hospital in the prologue of the second game but clings to life. As Max puts it, he's too stubborn to die from something like that.
Even more so in the sequel. After surviving the first game Isaac gets to spend a few years in an insane asylum. Then after being woken up to the sound of screams and blaring alarms, has his supposed rescuer killed and turned into a necromorph right in front of him. What does Isaac do? He HEADBUTTS that necromorph without hesitation and escapes, all while IN A STRAIGHT-JACKET.
Commander Shepard and their crew from Mass Effect. They face incredible odds to say the least, where failure is not an option and come out on top. Shepard and their team's reputation for We Do the Impossible is very well deserved.
Most of the alien races seem to think that humans are like this on a species level, largely because humanity's first contact with alien life came in the form of a short war; where despite only having mass effect technology for less than ten years at the time, they somehow managed to hold their own against the Turians, who had been the galaxy's reigning Badass Army for over 2000 years. The result was a 3 month long ground war that was locked in a virtual stalemate and the Turians gearing up for a full-scale assault on Earth, before the Citadel Council intervened and called a ceasefire.
Death itself couldn't stop Commander Shepard from killing Reapers. Garrus notes in Mass Effect 2 that the Collectors actually did succeed in killing Shepard, and that it only seems to have pissed him/her off.
Specialist Traynor calls him/her an "unstoppable juggernaut of headbutting destruction".
Zaeed survived getting shot in the head. "Rage is a hell of an anesthetic."
Tela Vasir of the "Shadow Broker" DLC is this in spades, She takes on Shepard tackling her out of a 4+ story building, survives massive car crash, loses tons of blood, and still has enough in her to be one of the toughest (if not the toughest) boss fights in the series.Bad Ass.
Kai Leng. In Mass Effect: Retribution, he escapes Anderson despite having TWO BLOWN OUT KNEECAPS by climbing a ladder with his arms and jumping onto his shuttle with said blown out kneecaps. In one trailer for Mass Effect 3, he is literally climbing on top of a speeding flying car in order to take down Shepard.
Thane Krios, a terminally ill Drell assassin. In Mass Effect 3, his Doctor's had given him only 3 months to live... over 9 months ago. Furthermore, despite his illness causing him difficulty breathing, he can still manage to come to Shepard's aid during the Cerberus Coup of the Citadel and kick Kai Leng's ass, preventing him from assassinating the Salarian Councillor. Unfortunately, Kai Leng manages to mortally wound him during this, but nonetheless, he dies a hero.
Shepard deserves special mention for the ending of Mass Effect 3. S/He's been shot by Harbinger's main cannon, which was designed to destroy Dreadnoughts, most of his/her armor burned away by the blast, and s/he gets back up and keeps on limping towards the Catalyst while blasting his/her way through Husks and Marauders.
It's also heavily implied that s/he's under greater stress in his/her resting state than s/he's ever been in his/her life, but tries to pass it off as normal battle adrenaline and keeps on going (almost) like normal. Keep in mind, that depending on Shepard's possible backstory, this includes almost single-handedly holding off 10,000 Batarians (War Hero), witnessing the death of their entire squad by Thresher Maws (Sole Survivor) or the Alliance's retaliatory strike against the Batarians at Torfan (Ruthless).
Krogans are this by nature. Mass Effect 3 reveals that the Graal Spike Thrower, a Krogan designed shotgun, was designed specifically for hunting Thresher Maws, towering high beasts which typically require a tank to take them down (unless they take the tank down first). Said shotgun was thus designed with special blades built in, meant to allow an unlucky Krogan swallowed by a Maw to either cut his way out, or at least cause severe internal bleeding to try and take it with them. Overall, it says something about the Krogan that they designed a gun believing it a very real possibility that they might be eaten at some point.
Dark Souls: Pretty much the whole game is an example of this trope, you will get crushed, hacked, beaten, roasted, poisoned, dropped down bottomless pits, eaten by hungry treasure chests and wanted dead by every single living thing in the game, it's strongly implied that even your friends will eventually turn insane and try to kill you. (Most of them do). Despite this, your character will keep pushing on and eventually kill every single badass in the game; some of which are essential gods, all in an attempt to try and remove the curse that has everybody going feral in the first place:. By the end of it all, it's not only the character that's a determinator, it's the player as well.
Prince of Persia: Warrior Within has the Dahaka, an embodiment of fate and guardian of the timeline, who hunts the Prince after he attracted its attention by his liberal use of time-altering powers that allowed him to cheat death many times over in the first game. And in the other corner the Prince himself, who is just as determined to find a way to get rid of the monster as the Dahaka is to kill him. A secondary character lampshades this:
Kaileena: I had hoped the Dahaka would kill you. I had hoped that Shahdee would keep you from the Island. I even cursed the sword I gave you, AND YET YOU DID NOT DIE!
Jon Irenicus of Shadows of Amn has one thing on his mind: the power he was denied when the Seldarine prevented him from becoming a god. If he has to rip the souls from living men and women, temporarily surrender to the authorities just so their guard is down and he can break out, work with thieves and vampires, make pacts with sworn foes of his people, pretend to be a humble servant, slaughter his kin or destroy entire cities to attain it, he will. You have to fight him no less than three times in the game, killing him twice. And after all that, after following him to Hell and kicking his astral butt there, the ending cutscene shows his soul incarnating again and attempting to fight off a horde of gibbering demons with his bare hands.
Sarevok, the Big Bad from the first game. He appears in the Very Definitely Final Dungeon in Shadows of Amn, and after dying there he appears to the player in Throne Of Bhaal, the final installment, having pulled himself back into a ghostly existence through sheer willpower in order to bargain his (vital) knowledge for the piece of the player's soul he needs to resurrect himself.
Sarevok;"Hahahahaaaa! I Live! Flesh and blood and bone! I swore I would scratch and crawl my way back into the world of the living... and I have done it!"
Crono of Chrono Trigger is pretty determined to protect his friends. "What's that, ancient evil? You've kicked my ass without even blinking and now you think you've won? Well, I'm not done yet!" (Then he's done.)
Magus is also a very good example. He survived in a (to him) post apocalyptic world, showering warfare on the world, all to find a way to kill the monster that ruined his life. His most famous quote embodies this trope:
Magus: If history is to change, let it change! If the world is to be destroyed, so be it! If my fate is to be destroyed... I must simply laugh!! I'm coming, Lavos!
Ayla is this trope to a T, in fact it's her whole life's philosophy and probably why there's a human race at all in the game world. Her most famous quote says it all:
Ayla: Ayla alive, Ayla fight! Win, live. Lose, die. That rule. No can change rule.
Alwan in the Geneforge series. After surviving an attack on his school, his dedication to the Shapers results in him becoming the General in charge of defeating the Rebellion, personally leading a stealth attack on the Rebellion stronghold, somehow surviving massive injuries after the attack goes awry, and becoming the leader of a faction despite ending up like this◊.
F.E.A.R.'s Alma is one of these, to a point where her sheer force of will allowed her to survive six days of being drowned with no life support (with one character commenting that she "simply refused to die") and her psychic presence continued to survive well after she finally died.
Unfortunately for Beckett, he found out that the only thing Alma wanted in the second game was to make him her man and have his baby.
The Replica troops are another variation on this, in that they will carry out whatever mission is given to them by their psychic commanders, immediately, without question, and without stopping until it has been completed, with no regard for their own lives. They'll go as far as suicidal frontal assaults to simply wound or slow down an opponent, to the point where they'll attack an enemy in a Humongous Mecha with rifles just to distract and delay it for a couple seconds.
Asagi from various Nippon Ichi games. Five games, five years, as many defeats by actual main characters, playing second banana to a freakin' Prinny, and all of it still has not extinguished her desire to, finally, one day, become a main character.
Not only that, but the protagonist also hangs onto life for two months after that, continuing on nothing but willpower in order to keep the promise the main characters made to meet on the roof of the school on graduation day as proof of everything they went through together.
All of the protagonists of Lunar: Eternal Blue fits this to some degree, except for Lucia. In fact, one of the key points of the game is the "power of humanity" to never give up even in the face of impossible odds. The best example is when the entire party is struck down by Ghaleon... and then, through sheer willpower, all five stand back up on their own and fight him a second time.
The Willpower defense powerset in City of Heroes is described as nothing supernatural, unlike Invulnerability, Regeneration, Fire/Ice/Stone Armor, and the like; the character's defenses stem entirely from the fact that even though "bullets don't bounce off of you, and if you are cut, you bleed," the character is "tough, grizzled and strong willed. It takes more than a little cut to keep you down!" Some examples of powers within this set are High Pain Tolerance, Mind Over Body, and Indomitable Will; the Eleventh Hour Superpower is called Strength of Will. Ironically, this is considered one of the better defense sets in the game, and is a favorite of Natural-origin Badass Normals.
Takeshi in Ever17 comes down with Tief Blau, hasn't eaten in over a day, has been running around constantly when not passed out, is coughing up blood and suffers some really nasty decompression sickness right after being slammed into a wall. Then he swims through frigid, lightless water to rescue his love interest, swims back still without having rested and finally escapes. Then he's jettisoned out of a submarine to save his love interest again and sits on the ocean floor for a while, having drowned. And then a disembodied time traveler wakes him up, at which point he swims through the bottom of the ocean back into IBF, saves Coco and still lives to the story's True End. Oh, and he was dead/frozen on the ocean floor for at least long enough for a small robotic dog to swim down and recover the disc he was carrying. Three hundred feet down.
In Knights of the Old Republic 2, this was Atton Rand's special ability. Unless he was the last man in the party standing, he literally got back up and kept fighting, his saving throws getting better the more damage was dished out on him!
Darth Sion, who has been shot, stabbed and blown up so many times that his sheer hatred and determination are literally all that's holding his body together. The only way to kill him is to convince him to give up.
Darth Sion: You seek to erode my will. You will not succeed.
In Chzo Mythos, Trilby qualifies for this trope, to the point where he is lying mortally injured, waiting for himself to be sacrificed to summon the avatar of a godly being and he Still. Won't. Die. Even when he acknowledges that it was only "his own stubbornness keeping him alive" and knowing that, should the Tall Man have no living sacrifice he would not answer the summons, you (the player) actually have to give Trilby the command to die. But then the ceremony still goes on, it just turns out that the Tall Man isn't picky with who he takes, instead absconding with the would-be sacrificer.
Subverted in 6 Days A Sacrifice, wherein we find clones of Trilby who are close enough to qualify as the real deal as far as Chzo / DeFoe / The Tall Man cares. Said clones are easily brainwashed, to the point where they serve as antagonists for half of the game. At the end Trilby (clone?) is seen quivering in terror inside of Chzo, cursed to the kind of immortal suffering that made the Tall Man into the Tall Man in the first place. Patently un-Determinator-like behavior all around.
The entire Ur-Quan race in Star Control II. After being forced to conquer the known universe and kill their only friends by their telepathic masters, an Ur-Quan scientist discovered that intense pain causes the telepathic link to shut down, supposedly to prevent their enslavers from experiencing discomfort. He (it?) proceeds to drink enough caustic acid to ensure its own agonizing death and takes the few seconds it bought with its life to broadcast its findings to all the Ur-Quan in range. The self-mutilation that followed bought the Ur-Quan time to develop a device that provided its user with constant border-line unbearable pain without inflicting physical damage. They proceeded to fight an entire interstellar war, constantly in throes of agony. No wonder they want to enslave/destroy the universe.
Metroid's Samus Aran. Certain logbook entries written by the space pirates in the Prime series mentions that no matter what they do, nothing has been able to stop the "hunter clad in metal".
Among the things they have tried are elite Commandos and Commanders trained specifically to kill her and only her, Elite Pirates with her common tactics programmed directly into their brains, turning her weapons against her, blowing up the planet she is on, and throwing Dark Samus at her, who herself has tried corrupting Samus with radioactive physics breaking phazon, collapsing a dimension on her, and blowing up yet another planet in an attempt to kill Samus. By this point, the Space Pirates truly believe that "The Hunter" is an Eldritch Abomination that has cursed them for eternity. And then, there's the sheer number of other random bosses and waves and waves of mooks. And then there's Dark Aether, where even the planet's AIR is trying to kill you.
Her arch nemesis, Ridley, counts as well. He was blown up before the series began, but recovered, then he was beaten in the original. He was turned into the cyborg Meta Ridley. The fact that this cyborg form looks entirely mechanical is a testament both to Samus' fury to inflict that much damage, and Ridley's tenacity to survive. This form is fought in Prime, culminating in him taking several lasers to the chest from Chozo statues, falling off a cliff, and exploding. He comes back in Corruption, first on Norion, where Samus sends him plummeting into a reactor core, then in a Phazon-corrupted form on the Pirate Homeworld, where she blows him up with a Phazon overload. He still returns, fully organic, in Super Metroid, where not only is he beaten, but the planet he's on blows up. In Other M, he's basically ressurected as a baby, escapes from his cage by killing the scientists who saw him as a pet, tricks a clueless Samus into helping him to eat and evolve, and as soon as he matured enough he proceeds to attack Samus. Said attack lead her to have a PTSD reaction, since she thought he was gone for good since Super Metroid. After regaining her senses, she manages to harm him enough to make him flee. While healing his wounds, Ridley notices an ennemy lurking in the shadows, and starts to panic when he realizes that his opponent is a massive Metroid Queen. She kills him by sucking all his life energy, leaving only his dried-out carcass behind. In Fusion, said carcass is preserved in a cold room, but is infected with X-Parasites before Samus arrived, and later shattered without her intervention, the X-parasites having assimilated his DNA. Later, Samus ends up fighting a X-parasite copy of a fully regenerated Ridley, who is once again defeated. After that, nobody knows how he's going to come back, but one thing is sure : Ridley just won't stay dead.
Crocomire. Anything that keeps trying to kill you even after being reduced to a skeleton should certainly qualify, even if it does just fall over dead without doing anything.
Left 4 Dead: The Tank, the hyper-steroid zombie juggernaut. He will take several shotgun shots to the head, get burned, or even scale a building to try to kill those survivors.
The Witch, who can do the same things as the Tank (except the scale a building part), all the while being more difficult to get rid of.
And now, Bill. The latest comic goes out of its way to prove this. He gets punched by a tank after dashing off to save his friends. The next panel shows him sat by a turbine, preparing to fight 3 tanks. And in "The Passing" DLC, you find him... sat in the EXACT same spot. Dead from blood-loss, but still in the exact same position.
Nick of Left 4 Dead 2 also counts, as shown by one of his more awesome lines:
"I have NOT come this far, to die now!
The Nemesis of Resident Evil 3: Nemesis fame should be the poster boy of this trope. It's a massive 8 foot tall B.O.W. that's programmed to kill all the members of S.T.A.R.S. after killing Brad right in front of you, it sets it's sights on Jill: The character who you play as, AND who happens to be the last member of S.T.A.R.S. No matter how far you run, or how many times you actually manage to put him down (About 15 shotgun shells on EASY will do the trick), he'll be back. Often with a ROCKET LAUNCHER. Did I mention it can run FASTER than you? Or that it's SMART?
T-101 from Resident Evil 2 also counts. While not nearly as memorable as the Nemesis (You only encounter him in a New Game+), he's every bit as unstoppable. Even dropping him in a vat of molten steel only manages to piss him off, as well as burn off the coat that was stopping him from going One-Winged Angel.
Resident Evil 6 gives us the Ustanak: a Nemesis Expy that haunts Sherry and Jake throughout their entire chapter. Like the T-101, dropping him in lava is only enough to seriously piss him off (And reveal his weak point, thankfully).
In Ōkami, Waka shows just how determined he can be just shortly after revealing he was Good All Along. First, he tries to fight Yami, until he loses his laser sword Pillow Talk.Then when Yami starts dishing out Death Rays aimed at Ammy, he blocks them with his regular sword. And even once he is completely disarmed, he still stands in front of ammy to take one last attack for her.
Hakumen from BlazBlue spent ninety years of utter isolation in the Boundary - the same place that drove Lotte Carmine mad and turned him into the Eldritch Abomination we know as Arakune - and retained his sanity through sheer force of will.
Also LitchiFaye-Ling. There's nothing stopping her from attempting to find the cure both for Arakune and her, be it the Boundary's memory erosion, nearly everyone else's suggestions that it's a lost cause or even desperate decisions to join Team Evil. This is also probably why similar to Makoto and Hakumen, she didn't devolve into a puddle yet (though she's only getting slow symptoms simply by being a mere human).
Shu from Blue Dragon is this, through a combination of youthful exuberance and steadfast...um...determination to protect those he cares about.
The Hashashin from Assassin's Creed. With all of the conspiracy theories and rewritten history strewn throughout the two games, only one thing is left absolutely certain: if Al-Mualim sends Altaïr after you, there is a 100% probability that you are going to die.
You also do not want to kill most of Ezio's family, if you don't want your entire conspiracy to be ground to dust by a single man, whose Roaring Rampage of Revenge ends with administering a Curb-Stomp Battle to the Pope.
Revelations just turns this Up to Eleven: Ezio is 55 years old, has been fighting and training most of his life, and is the Mentor of the Assassin Brotherhood. One would think that he would be slowing down in his old age. But as the intro movie shows, he not only hasn't slowed down, he literally hasn't missed a beat. The only reason he's defeated by the Templar soldiers is because he catches a glimpse of Altair in the melee, which distracts him just long enough. And then, after he's been defeated and is about to be hung...well, let's just say that all he's doing is waiting for the right moment to make his move, which is just as the noose is tightened around his neck.
In Assassin's Creed 3, Connor is just as much this as Altair and Ezio. He is betrayed by Washington and the Patriots after all the help he gave them, is shellshocked by cannons that he himself arranged to be fired at the fort he was going into, is nearly strangled to death by his own father, and ends up being impaled by a broken beam in a burning ship. When Charles LeeLampshades this and asks why he goes on, Connor replies "Because no one else will" and shoots him, before proceeding to hunt him down across the colonies.
One of the bosses in World of Warcraft (Golemagg, in Molten Core) involves two giant dogs which are pets of the main boss. If you try to kill them while Golemagg is still alive, the game prints a message saying "Core Rager refuses to die while its master is in trouble!" and they heal back to full health.
Holy priests and protection paladins can both become these. Ardent Defender reduces the amount of damage the paladin tales when at low health, and when causes a blow that would otherwise kill them to instead heal them again (this part has a cooldown period). The priest talent Spirit of Redemption allows them to stay on as a ghost after death for a short while. Though they can only cast healing spells in this period, meaning that they're staying on long enough to see other people kill their enemy.
A properly built high level tanker in City of Heroes is pretty much unstoppable unless you do something very stupid. Only giant monsters and archvillains should really be a challenge.
Phoenix Wright in the Ace Attorney series. No matter how much the odds are stacked against him, Phoenix refuses to give up and if he is on the verge of doing so, he'll meet someone who will convince him otherwise. On the same token, except for Winston Payne, almost every prosectuor and witness refuses to give in, no matter how much evidence and logic the defense throws at them.
One could arguably give this title to Dahlia Hawthorne, given that she won't even let death stop her from seeking her revenge on Mia Fey.
The villian you confront in the final case of Miles Edgeworth: AAI. Quercus Alba, an ambassador, is the kingpin of a smuggling ring and has ties with the KG-8 incident and its second occurance (case 4). Edgeworth tries to prove that Alba was the one who killed Manny Coachen, but no matter how much logic and evidence Edgeworth throws his way, Alba never gives in and keeps demanding more proof. The entire last half of the case is devoted to just cross examining Alba until he finally cracks.
Manfred von Karma needs more recognition than that. He took a bullet wound to the shoulder, and left it. He was afraid that if a doctor got to it, that doctor would one day testify against him. All of this so he could get his revenge on the Edgeworth family. And also, he retrained a parrot. Just in case someone would be daring enough to have it testify.
Boomerang from the first Wild ARMs. He betrayed and hunted his own race simply for the thrill of the hunt, killed one of his (possible) brothers when he saw the humans were more satisfying prey, is betrayed, attacked, and killed by another brother (even though Zeikfried is, at the time, possessed by Mother), and then, simply because his thirst for the hunt hasn't been satisfied (because Zeikfried/Mother interrupted his climactic battle against Rudy, Jack, and Cecilia), he fights his way out of Hell itself so he can finally fight the heroes and have his perfect, final battle.
Boomerang: Miracles, the humans' belief system that makes the impossible possible. The power that protects Filgaia. Looks like this is the end. Did I lose? No, I did not, I lived the life of a Demon Warrior! I fought and lost my life, I was not defeated, I lived a life of my own dreams. I have lived, I have no regrets, I am a winner. My death is glorious.
The guards from The Elder Scrolls IV: Oblivion fit this trope perfectly. Because they are always ten levels higher then the player, they can take a beating that would slaughter an entire family of bears. They will chase you across the entire country, through rivers and over mountains, passing bandit camps and hell gates swarming with demons, until they finally manage to corner you and perforate every one of your internal organs, because you touched a loaf of bread. Not stole, touched.
As shown in this video, never, EVER sit in Janus Hassildor's throne!
Try playing a round or two in Team Fortress 2 against a competent and well supported Heavy. Between ungodly amounts of health and a minigun the size of a refrigerator, a good Heavy can just keep plowing through enemies as long as he's got the ammo, and sometimes not even that's necessary. There's even an achievement for the Heavy where he has to absorb every discreet type of damage the game has to offer (bullets, explosives, melee attacks, fire) and still keep going. Another requires the Heavy to survive a crit rocket, one of the most devastating types of Critical Hit in the game that would turn every other class into a confetti of body parts. Yet another demands tanking 1000 points of damage in a single life, or 333% the Heavy's normal HP stock. Definitely a class where pure dogged relentlessness is rewarded.
The Soldier's Equalizer allows him to qualify. The less health he has, the faster he runs with it out and the more damage he does - at less than 25% health, he can kill a Heavy in a few non-critical hits. Of course, that requires not charging into the Cone of Death.
Of all the innumerable charaters from this series who could be put on this page, Aqua should be here. Ten years of living in the dark realm without ANY protection, and she hasn't given in to the darkness. NOBODY is supposed to be able to do that, period, for more than a few minutes. Aqua does it for 10 years. As of 3D, it's been twelve years.
Cobra from the Silent Scope series. "I am immortal! Even if my body collapses, I will not die until I defeat you!"
The main character of Zettai Hero Project is a shining example. Cheerfully called "the weakest main character", he was just an ordinary bystander on the way to some errands when the mantle of The Absolute Victory Unlosing Ranger, the world's most popular super hero was passed on to him. He then proceeds to the Final Boss. Naturally he gets nearly killed. After a bit of training, he does it again. He fails. Repeatedly. He gets mocked, he gets jeered, and nobody believes in him. They call him a loser underdog, they say he looks stupid, and they accuse him of being a fake. But in the face of insurmountable odds, in the face of staggering disbelief, in spite of intense pain and injuries, in spite of the fact that this was forced on to him, he keeps on getting up. He keeps on fighting, getting stronger bit by bit, inch by by painful inch, until he can finally save the world from Darkdeath Evilman. The crushing responsibility of saving 6 billion people rests on his shoulders and he keeps on carrying that weight forward. Too stupid to give up, too stubborn to lose, The Absolute Victory Unlosing Ranger.
In fact, we learn later that Main Character's Establishing Character Moment happened when he was just a little kid without any superpowers—he and his sister were kidnapped by a Serial Killer. Despite getting beaten to a near pulp, Main Character just kept standing back up over and over and over again until the police arrived.
The titular Super Meat Boy. Razor-sharp saws, piles of needles, rivers of salt, and molten lava will not stop the guy from rushing in to stop Dr. Fetus and save Bandage Girl, who is always right before his eyes before Dr. Fetus comes and kicks her away to the next level. There is absolutely no stopping Meat Boy.
Poppy from League of Legends as a child witnessed her father (a blacksmith) being assassinated as his caravan was delivering a "glorious helm" to a famous general. She escaped with the helm and delivered it on foot and alone. In game, her intrinsic effect is to reduce all damage over 10% of her current health by half, making her harder and harder to hurt the more damage she's taken. One of her abilities is to gain damage and armour each time she attacks or is attacked and another renders her completely invulnerable to all enemies except for one of her choosing for 6-8 seconds. Her role in a team fight is generally to make it through the melee and take down the most dangerous enemy champion.
Poppy has nothing on Nautilus. He was dragged to the bottom of the sea (bringing the anchor of his ship with him). His diver's suit fused to his body and his memory was fuzzy at best. Despite all of this, he walked along the bottom of the sea, in complete darkness and loneliness, to find the people who left him to die and make them pay. In-game, he's a Mighty Glacier that can take a lot of punishment before going down and has some respectable damage behind him.
Garen's passive ties in with his hometowns military rules of not retreating, making him a determinator.
Alcatraz in Crysis 2. He essentially became a corpse at the start of the game (and you'll get a more comprehensive explanation why a bit into the game), only kept alive by the Nanosuit Prophet puts on him. And even after that, he goes through a lot more punishment that should have put him down even if that suit was on Maximum Armor. One of the more significant injuries he sustains is, after escaping the Nexus, he gets knocked out when a car falls on his head, and then he drifts down river until he wakes up to Gould and Tara Strickland and just gets into an IFV and soldiers on.
A tie-in comic to Portal 2 reveals that Chell, the player character, was actually to be rejected from testing for her off-the-scale, nearly pathological tenacity.note It may also count as a bit of a meta-joke on her being the Player Character and therefore unstoppable in the face of infinite reloads. A psychotic researcher arranged for her file to be transferred to the testing area because she had the right stuff to take on GLaDOS, and the rest is history. This trait is Lampshaded by the Big Bad after she's survived a Booby Trap that, by all rights, should have killed her.
In Portal 2, this is shown to extend beyond Chell and even to things she creates. An area of the facility where a sign explains it was once the location of a "Bring Your Daughter To Work Day", implies that Chell is the daughter of an Aperture Employee and once took part in a science fair, where they were tasked with creating a simple potato battery. In the many years since, while the other experiments have sat gathering dust, Chell's potato instead grew out-of-control and even sprouts through the ceiling!
Said researcher himself counts as well. The sole surviving scientist of Aperture Science and GLaDOS' wrath, he managed to elude her for years and scribble maniacal ravings within hidden chambers with no portal device, only the "help" of the Companion Cube and his own schizophrenia. Upon seeing Chell being dragged back into Aperture Science, rather than escaping he goes back to save her and ensure that she survives until Portal 2.
Cave Johnson also counts, refusing to let government investigations, years of failure, bankruptcy, and his own staggering incompetence stop him from doing science!
The Courier in Fallout: New Vegas. You take two bullets to the head and get buried alive in the opening cutscene, and revenge is the MacGuffin for the first part of the game. One quest has you confront the goons of the guy who shot you (however, you can actually help them get out of the mess they've made for themselves), and another has you confront the guy who pulled the trigger. Meanwhile, as you're doing this, you're bound to incur the wrath of at least one ill-tempered fighting force at some point in the game. Caesar even comments on your tenacity when you first meet him.
Similarly, in an unmarked quest in the game, a woman asks you to recover the body of her soldier husband from Fiends. If you go past the soldiers holding the line a ways away from the body, then come back with little health left, the soldiers comment on how you're barely holding together and wonder aloud why you did this for someone you don't even know. One of the possible responses is, "I made a promise to Private Morales." The soldier states that he'd hate to be between you and your goals.
In Chaos Rings Omega, Olgar/Dante arrives in time to rescue his son-in-law Vieg (soon to become Olgar) from Yorath. He immediately gets hit by what should be an instant-kill, but then continues in a scripted battle where Olgar can never lose his last hit point. And then, when all reason suggests that Olgar, who just defeated someone who could defeat the Qualia, should be deader than dead, saves Vahti and her child Ohm by jumping into LAVA. He walks through it for what seems like hours, telling Vieg to take over protecting the family from him. Finally, when he carefully lays his daughter and grandson next to the still nearly conscious Vieg, he finally sinks into the lava. Apparently willpower keeps your legs from being burned off.
Asura from Asura's Wrath is possibly the most over the top of the examples shown so far. The trailers and gameply demo have seen him get impaled by alot of spears and shrugging them off as if they were nothing, knockoff a demi god that grew in size to about Five stories tall even after being bashed into the ground, then said demi god grows to size where he dwarfs the fucking planet and send his finger (which alone makes every boss from every game ever look absolutely puny in comperison) down to crash him. Asura STOPS the finger with his bare FUCKING hands! (All six off them, yes he grew another two pairs of arms, roll with it), THEN he destroys the demigod by doing what basicly amounts to punching it's finger alot.
The newest trailer then shows Asura, his arms destroyed after punching the demigod's finger, being confronted with a figure named Yasha. We know nothing of Yasha besides the fact that he knew Asura once upon a time, and he pissed Asura off at one point. Asura, who I must remind you has no arms, attacks Yaksha with kicks and headbuts even though they apear to be of no effect. Yasha throws him to the ground boasting that Asura's attacks can't harm him. Asura isn't done yet. He charges forward, shouting out Yasha's name in rage, and headbuts straight in the face! Asura brings a whole new meaning to the word 'Determinator'.
Nathan Drake of Uncharted fame. The very first scene of the second game is Nate hanging from the end of a train car that's dangling over a freezing cliff in clothes that can not possibly be warm and a bullet in his stomach. He climbs all the way back up despite nearly every part of the train either falling on him or breaking when he tries to use it as a handhold, presumably because every piece really wants to be the one that finally kills the resident Chew Toy. He even notes that it would be easier to let go, but instead he gets all the way back up just so he can trek through miles of freezing snow while hopefully not bleeding to death.
Uncharted 3 almost makes this the centerpiece of Nate's character. After he's captured and tortured, he breaks out of captivity, beats the crap out of his captors, and, rather than run away, fights through the rest of them to save Sully. When he finds out that Sully isn't even there, he (accidentally) sinks the ship that he's only, barely making it out before dying. He then drifts to shore, barely alive, and manages to find Elena, who tells him that there's a plane getting ready to leave and meet the convoy where he'll find Sully. Despite barely being coherent, he turns around to go find that plane.
And to top it all off, Nate then survives for three days in the Rub 'Al Khali desert, despite no supplies, no water and no idea of where he's going, topping it off with a gunfight. The man literally cannot be stopped.
The Kid from I Wanna Be the Guy. Also, anybody who manages to play through I Wanna Be The Guy.
Cruiser Tetron, the Big Bad from Hero Core. Get him down to a sliver of his health and he'll collapse in the corner, unable to even stand, and only able to briefly lift his gun to fire at you. But he still keeps trying to fight.
Red from Solatorobo. A recurring characteristic of his is his willingness to see things through to the end, no matter what.
Many RPGs have an ability or statistic called "guts" that allows heroes who should by rights be dead/unconscious to stay up on 1HP through sheer willpower. See Last Chance Hit Point for examples.
Billotte's Medal in World of Tanks can only be earned by a determinator whose given his enemy a streak of Why Won't You Die? To get it, you have to survive 5 Critical Hits, have, at most, 20% of your health left, get one kill, and survive.
In Guild Wars, The Duke of Ascalon survives the apocalyptic events of the Searing, which utterly destroys Ascalon. But he doesn't stop there. He takes his army of soldiers and elite guards, and pushes into Charr territory, the creatures who unleashed the Searing on him in the first place. Then, he captures and secures the ruins of Piken Square, and creates a small TOWN there. THEN, he has to deal with the death of his daughter, the lovely Lady Althea, the Prince of Ascalon's girlfriend and fiancee. It only strengthens his resolve.
Guild Wars 2 takes place hundreds of years after Guild Wars. When playing from the Charr perspective, your first enemies are — of all people — the Duke's elite guards in ghost form. And their boss? Duke Barradin's ghost. It takes a sizable force of NPCs and players to take him down. Even after you destroy his ghost, he possesses a giant statue, and then it's ROUND TWO. And when you manage to destroy that, IT'S ONLY TEMPORARY. The Charr leaders say he'll just reform in a few weeks and attack again. He's long dead and hundreds of years have passed, but he clearly won't let minor details like that stand in the way of him defending his homeland from invaders. The man is a BadassDeterminator in every sense of the word.
There are several in Starcraft and Starcraft 2. Jim Raynor is the most obvious one who refuses to give up on saving Kerrigan and bringing down the Dominion even when logic and good sense says there's no way for him to win.
The Kid from Bastion. Even before he found himself thrust into an epic struggle to rebuild the world, he volunteered for two tours of duty on the Rippling Walls, a defensive barrier where life was so rough and dnagerous that no one in the history of Caelondia had ever volunteered for a second tour. Afterward, he hurls himself into saving the world, despite the fact that he is alone, vastly outnumbered by monsters and beasts and the heavily-armed remnants of the Ura, and armed with only scrounged weapons, techniques detailed in old, dusty tomes, and a whole lot of alcohol. It culminates at the end of the game, where the kid finds Zulf, a former friend-turned-instigator of the Ura's attack on the Bastion, who is being beaten by his fellow Ura for convincing them to attack the Bastion and losing so many lives fighting the Kid. If the Kid chooses to save Zulf, he picks him up and slings him over his back, abandoning the all-powerful Battering Ram, and walks through the assembled Ura army, taking wave after wave of crossbow bolts and pressing on. Eventually, the Ura are so amazed and impressed by his unwavering resolve that they stop attacking and stand aside...save for one jackass who opens fire on the Kid near the end, and is immediately cut down by his commander.
Fujiwara no Mokou from Touhou. She's a normal human apart from the fact she's A. immortal (via resurrection), B. can control fire, and C. fly. Despite her immortality she is as squishy as any other human and feels the pain of her injuries. She is to date the only character to explicitly die on screen in a game. The only thing that stopped her was dying enough times where the pain was so unbearable she could no longer focus on the task at hand.
Marisa also qualifies as well, always pushing herself to be as good as Reimu.
Little Busters! has Kyousuke. He reset the world countless times searching for the one way to make Riki and Rin strong enough to survive without him, and even when one of his plans turned out disastrously and he sunk into guilt and despair, he still never gave up. On top of that, every night while he slept he would replay the scene of the crash, crawling on his stomach through darkness in an attempt to reach the oil leak before it catches fire and explodes. When he finally reaches it and realises that when he wakes up he'll be reset back to his starting position? Well, he just needs to change his starting position. How? Well, his starting position is based on when he died, so naturally he has to kill himself. So he does so without hesitation. It works. Also, Kyousuke's plan was to turn Riki into one of these. It also works, and by the end of Refrain, Riki will do anything necessary to help his friends.
Wario from pretty much any game he's in (Wario Land or WarioWare). The best examples here are the second and third Wario Land game, where he physically is unable to die in any way. That's it, nothing except the third game's final boss does damage period. Fire just makes him run around and break blocks, zombification doesn't actually kill him and being smashed flat is a beneficial status. Hell, Wario World literally has him go One-Man Army throughout the game from start to finish.
Once the titular detective from the Carol Reed Mysteries has set her mind to solve a case, she won't stop until the mystery is unraveled. In one game, she even forgets the date with her boyfriend on Midsummer's Eve she's been looking forward to for weeks because she's so wrapped up in a case.
Mega Man Legends series: The Bonne family could be considered this. Throughout the series they are constantly beaten in battle by some blue kid and his friends, or find themselves completely out of money, but they don't give up. It's even noted by an NPC partway through game 1. Through all of the explosions and failures, the Bonne pirates always find a way to survive and attack Megaman again and again. (If only Capcom shared their determination...)
Joel and Ellie from The Last of Us. They spend the better part of a year going back and forth across a ruined North America trying to find La Résistance. All the while not only having to find supplies to survive but also fight off hordes of human and infected enemies as well. Heck, even near fatal injuries don't slow Joel down after a shot of Penicillin.
Lee Everett in The Walking Dead in the final chapter. Being infected, being alone, a street full of about 80 walkers, and having only one arm isn't enough to stop him from rescuing Clementine.
Dynasty Warriors 7 tragically deconstructs this term, especially for the Shu side. Jiang Wei becomes so obsessed with fulfilling his promise to Zhuge Liang that he kept attacking Jin despite failing everytime, draining the kingdom's resources needlessly and getting many soldiers die. His determination to fulfill the promise to Zhuge Liang is that big that he is unable to listen to anything else. When Liu Shan eventually surrenders to save the people of Shu from the Hopeless War he created, Jiang Wei still refused to go down quietly and joins up with Zhong Hui for a chance to fulfill that promise, resulting his death.
Armageddon MUD turns the entire dwarven race into a collective bunch of determinators. In the course of growing up, every dwarf acquires a focus. A focus is, essentially, an extremely difficult goal which will tend to requiring a lifetime of effort, struggle and work to acquire/maintain. In the event that a dwarf does somehow manage to complete his focus, they will simply set themselves a new one, one often related to the first, and usually much tougher and difficult still. There is nothing at all for a dwarf to stop in doing their focus; a dwarf will completely consider any detail that might throw them off course and will not stop until the outcome of said focus is assured and finalised completely.
Dwarf Fortress dwarves tend to stand out among other races for this. While goblins run the hell away when their leader falls, and elves and humans are quick to fold them when they see everything's going belly up, you seldom see a military dwarf run away from anything. Chop off their arms and they kick you, chop their legs too and they'll bite your face off, but they'll die in battle, even if it stops being so heroic and starts crossing over into stupid. And once the threat's gone, they'll gladly get back to work while missing half/all their limbs, or have a spinal injury in a game without wheelchairs, or went completely blind, or are dragging their intestines behind them. Or all of these, depending on how cruel the Random Number God/the player is to that particular dwarf.
Nothing, and I mean nothing, will stop the protagonists of Silent Hill from their goals, except maybe getting abducted by aliens.
Clive Barker's Undying: Let's face it: Patrick pretty much IS this trope. Let's see here: Survive World War One? Check. Survive at least decades of conflict with Otto Keisinger? Check. Get through Irish customs unnoticed in order to fulfill a life debt? Check. Fight through wave after wave of unspeakable abominations that have racked up quite the kill count? Check. Handle the Artifact of Doom with fairly marginal damage? Check. Kill off the undead and superhuman Covenant siblings one by one? Check. Go into HELL to defeat Keisinger? Check. Survive Jeremiah's betrayal? Check. Take down the Eldritch Abomination that helped cause this godforsaken train wreck in the first place? Check. Basically, by the end, he is pretty much surviving more-or-less because it seems like he can't bloody DIE.
While all the characters in 1001 Spikes can technically be seen as this, it's the main character who plays it to a tee. His whole motivation is to prove himself to his rather emotionally abusive dead father, and no amount of spikes, fire, poison darts or any other sort of death will stop him. Taken up to eleven in the second part of the game, where, after he hears his father might not be dead at all, he goes to explore Antarctica with the sole goal of meeting him again and punching him in the face.