Dungeons & Dragons actually has monsters based around this trope. The Inevitables, a race of extra-planar automotons, are designed to uphold the universal concept of law. Someone who has escaped a fate destined for them, or otherwise gained the ire of the Inevitables will be hunted down unceasingly by these creatures. If you manage to kill one that's after you, they'll just send more. No matter where you hide, no matter which plane of existence you flee to, they will hunt you down until you are dead.
Characters that take the 3.5 Diehard feat (and its required precursor Endurance) tend to be the ones that will finish a fight, even if it kills them. (D&D has a 10 HP buffer between KO'ed and dead as a doornail; Diehard allows a character to ignore the KO and keep fighting at the cost of their few remaining HP.)
Ditto druids who take the Boar's Ferocity feat (Complete Divine). They can spend a Wild Shape use to keep fighting for ten rounds after going below 0 HP.
The real Game Breaker is the Frenzied Berserker class. When one of these goes into his Unstoppable Rage, she literally cannot be killed or stopped by hit point damage. The frenzied berserk in question may have less than 100 hit points, but you can do millions of points of damage and she'll continue fighting. This state of affairs will continues until she calms down, at which point her wounds suddenly take full effect.
However, in the rules, they specifically mention that non-hit-point-damage ways of killing still take full effect (such as being brought to -10 hp by a disintegration spell or by a Death effect).
Some animals in the game, such as badgers and wild boars (as well as their giant-sized monstrous counterparts) are given an ability similar to the Frenzied Berserker.
4th Edition's Epic Destinies give almost any epic-level character a means to cheat death. The best of all might be the Undying Warrior destiny, which allows the character to will himself back to life within seconds. The only drawback is that it takes longer to do each time you die in a day. At most an Undying Warrior's enemies will get a single day to run away before he comes back.
The previously mentioned Diehard returns in 4th edition as an Endurance skill power and allows a character to continue fighting below zero HP at the cost of being dazed.
Crusaders have a class feature called Steely Resolve, which allows them to ignore a certain amount of damage until the next turn, allowing them to take measures to mitigate the damage before it actually applies. They also gain bonuses to their attacks while suffering from such delayed damage. Eventually they can learn a combat stance that allows them to ignore damage that would knock them unconscious, if they can make the appropriate saving throw.
Arcana Unearthed/Arcana Evolved had the player character class Oathsworn, whose abilities, such as "deny hunger," "deny sleep," & "deny fatigue" were contingent on accomplishing something within a year (or preventing something from happening for that year).
The Necrons and their C'Tan gods are [un]living embodiments of this trope. Shoot them, they repair themselves. Blow them to bits with explosives, they reassemble. Rout them, they phase out and will come back for more eventually. Shoot the C'Tan with the combined power of thirteen Blackstone Fortresses, each capable of destroying a star (and the system along with it), and which fire the only weapon in existence, they're noted as powerless against and vulnerable to, and they take a nap (although how much damage the C'Tan in question sustained is unknown, as it has been "napping" ever since, for 60 million years, and doesn't show any indications of waking up yet). Force the entire race into hibernation, and a few billion years later, they'll wake back up and resume the attack with gusto. Each and every Necron is a Determinator, and there are thousands of them at minimum, complete with physics-bending weapons, vehicles, and starships.
Of the Sisters of Battle - fanatic, zealous warrior-nuns with flamethrowers whose faith is strong enough to stop artillery shells - the Order of our Martyred Lady are known for being [i]especially[/i] determined.
Earlier fluff told us about Ollanius Pius, a Guardsman that stood up to freaking Warmaster Horus himself. There is a reason he had become a patron saint of the Imperial Guard.
A more specific example would be the Death Korps of Krieg. These regiments are renowned as Determinators in an army of Determinators. The Death Korps soldiers unflinchingly advance under fire and dig in with World War I style tactics, seeing their massive casualties as penance for a planet-wide rebellion on Krieg millennia ago. They won't give an inch of ground as long as there are guardsmen living to grind it out in the trenches.
An even more specific example is Commissar Yarrick. Not particularly special, except that if he loses his last wound, he has a 66% chance of getting back up. Even if it was an anti-tank weapon, he can still get up and keep coming.
The Space Marines embody this trope to the t. Not only are they heavily armoured supersoldiers with the ability to withstand wounds that would kill a normal man thrice over, they are also unshaken n their faith and determination. Some notable examples include the Ultramarines 1st (veteran) company that defended their chapter fortress against the innumerable Tyranid swarms. They managed to hold them back long enough for the orbiting fleet to drive off the Tyranid hiveships, even though it cost them all their lives (as well as several nearly irreplaceable suits of Terminator armour). Also deserving of mention are the Grey Knights, which are always eliter-than-elite's-elite, who has never had any of its order succumb to the call of the Chaos Gods. Then you start going into the named and higher-ranked characters and see what Warhammer 40K is going for. It is said that "A fortress will not stop the Space Marines, although it may slow them down"
Relatively averted with the Tau, whose military doctrine considers ground won as irrelevant beyond being a place to kill the enemy. They consider a heroic Last Stand to be the mark of an incompetent or unimaginative commander, and won't hestitate in retreating from a strong enemy attack. However, that all goes out the window if a Tau army's Ethereal leader is killed. The Tau version of battle-rage involves slowly advancing while pouring ridiculous amounts of fire into the enemy, only stopping when they run out of ammunition.
Note that each and every Tau is absolutely willing to die for the cause of Greater Good. Even Ethereals should the situation call for it. They just don't like to spend their lives inefficiently. After all, they don't have quadrillions of people like Imperium does and their plot armor is lot thinner than Space Marines one.
The Orks have an... Interesting view about defeat. Basically, they never lose. If they win, that's good. If they're defeated, they're dead, so that doesn't count. And if they retreat, it's for coming back later fur annuver go with more boyz and More Dakka. Those green barbarians just won't stop coming at your throat until you'd have wiped them all from a planet's ground... And as they emit tons of spores upon their deaths which will form new Orks, even this won't stop them from coming back. Unless you Kill It with Fire, in which you may reduce it.
The Death Guard Leigon have Plague Marines, who simply laugh at the bullets and las-fire going straight through their rotted, decaying chests and continue to shamble towards the enemy.
Chaos Bikers with the Mark of Nurgle. They are quite literally immune to any bullet smaller than a boltgun shell.
Those declared Living Saints tend to have this in droves. Saint Celestine, the only one with in-game rules, gets to make a leadership test any time she suffers what should be a lethal injury. If she passes, she comes back. On average it would take twenty-six hits from a strength D weapon (read: anti-titan weapon) to keep her down. An Emperor Titan, a hundred metre tall walking cathedral/weapons platform protected by layer upon layer of energy fields, takes only twenty-four.
Sadly, this was reduced over the years. First her ability to stand back up was reduced to 50/50, though if she failed she would get another chance next turn instead. As of 2014 she once again rolls a leadership test, but can only stand back up once per game.
The Emperor himself, who despite having watched his Imperium tear itself apart, had to kill his favored son while most of the rest either died or turned to Chaos, and effectively dying at the end of it all, has held the Imperium together through his sheer psychic will for over 10,000 years.
Although things aren't as depressing as in Warhammer 40 000, Warhammer Fantasy Battle has its own share of Determinators. As far as determinator armies are concerned, the Dwarves, Orcs and Chaos warriors qualify best for the term.
It's also not wise to get between the Lizardmen and a goal, either. Don't try and take their sacred golden plaques, either, because they will literally hunt you to the ends of the earth to get them back, whatever the cost.
The Tomb Kings don't take theft lightly either, they will send an entire legion of skeletal warriors just to take back a single coin from them. The worst part is even if they are undead, and they will keep coming back from the dead no matter how many times they are killed just to retrieve their stolen property.
7th Sea's Erich Sieger, a ruler so stubborn that he salted the earth of his own lands to prevent an invading army from capturing it. He's the poster boy for the mechanical advantage the system calls, "Man of Will".
Shadowrun gives us the Dead Man's Trigger rule, which allows a character to make one final action when they would normally be dead or unconscious.
GURPS has quite a few Advantages that exemplify the Determinator. First off are Indomitable and Unfazeable: the first makes it so that no social influence (barring from those that take a special Advantage themselves, and even then you can take the Cosmic Enhancement to protect against it) affects your character, and the latter means the character can never be intimidated or scared. Also, you can purchase Will above 20. On the physical front, there's High Pain Threshold, Hard to Kill, Hard to Subdue, Supernatural Durability and all three versions of Unkillable: for the most part, these can be explained as heroic determination in the right campaigns. Fatigue Points can be purchased in bulk, representing "inner drive".
Many of the Exalted. To quote the sample Dawn Caste, Dace: "AS LONG AS SUNLIGHT BURNS IN MY SOUL, I'LL FIGHT ON!"
Two Charms in particular can help with this; the Solar charm Final Ray Of Light allows a Solar who's been killed to stand up again at half strength, possibly with Libera Me (From Hell) playing, while Infernals (when Ink Monkeys stuff is permitted) have Driven Beyond Death, allowing them to respond to being killed by activating a Shintai and continuing to fight until their Overdrive pool runs out - and with the right Malfeas charms, this can take a while, and if their enemies aren't around to finish them off when they do finally run out of juice, they'll recover.
While Sidereals Know When to Fold 'Em by definition, they can continue fighting on, and on and on, with the right platter of charms. Heartless Maiden Trance make a Sidereal immune to penalties from wounds, he doesn't get tired, he can't be harmed by the environment, and so on. Cue a DeterminatorNinja. Once he ends this charm, however, all the hurt he is supposed to suffer catches up with him. And he has to, because he can't regain Essence while this charm is active.
Werewolf: The Apocalypse has a rule that a werewolf who is killed rolls their Rage stat and heals one health level per success, because they are literally too angry to die.
Legend has a feat called "By Will Sustained". By level 12, you are so dedicated you no longer need to breathe.
This is actually statted in for the Gray Death Legion in BattleTech. They don't suffer from intimidation checks and gain several bonuses if outnumbered or if they have lost units, all in the name of perseverance and not quitting, as befitting their history. The fluff even says that "The only thing more dangerous than a cornered Gray Death Legionnaire is one in a damaged 'Mech." They even have a mechanic where their designated commander, if pinned down and unable to escape, can take their shot before anyone else can even pick a target so as to try and turn the tide.
Nobilis and Chuubo's Marvelous Wish-Granting Engine both have a rule that it's possible to use a Bond to overcome adverse circumstances, such as starvation, lack of sleep or even asphyxiation, as long as you are actively pursuing that Bond at the expense of everything else. Hilariously, in Nobilis this is mostly used for epic acts of lava-walking defiance, while Chuubo's mostly uses it to enable those "how long have you been hiding in my refrigerator" moments, but in the more dramatic stories it can be used in the same way.
Every single non-human being in Anathemahas to be this or the other determinators will destroy them.
Anyone in Infinity with the higher-level Valour skills, V: Dogged and V: No Wound Incapacitation. Dogged allows you to choose whether running out of Wounds knocks them properly unconscious or lets them continue acting for a little while before dropping dead. NWI goes so far as to permit them to just ignore being unconscious; unlike Dogged, this allows them to be healed, although taking another injury will still kill them.