Ichigo seems to need to get beaten to within an inch of his life before he can call on his true power. When the Big Bad tried to split him in half, the wound only went through the spine. Aizen commented that it was Ichigo's will to live that was keeping him conscious and alert when he should have been dead. At one point, he gets a hole blown in his chest, and it was stated to be unhealable. He. Got. Better.
Byakuya is willing to cut his own tendons and keep fighting with only one functioning arm and leg. He went through hell to both defend Rukia's execution from Ichigo and then protect her life from the Big Bad, due to conflicting vows he'd made to uphold the law and protect her life. When brainwashed into believing a villain was his mentor figure, Byakuya still resisted the brainwashing purely because it didn't change the fact the villain was an enemy of Ichigo's.
Kenpachi doesn't think death is an excuse for losing a fight. He lives by the code that anyone who wants to give up with an "honorable death," instead of fighting on to MAKE what you want happen, is just a whiny little bitch. It's eventually confirmed that he needs to be brought to the verge of death to grow stronger.
Yumichika doesn't use his zanpakutou's power in battle. He would rather die than have his division find out what he's capable of. As a result, he pretends to activate his power and then simply fights on sheer willpower alone.
Renji during his battle against Byakuya. After being cut over a thousand times by his opponent's Bankai and being cut again by some thrown swords, he still continues his battle and attacks him with his sword and reaches him. The blade of Renji's sword falls apart before it could pierce through Byakuya, but he still hits him with it, and then falls unconscious. Highly impressed by Renji's willpower, Byakuya appreciates this by throwing his incredibly expensive scarf at him to show his respect.
Rukia during her battle against Aaroniero. At the edge of death, being pierced through her chest by a trident doesn't stop her from killing him.
Grimmjow doesn't give up continue his battle against Ichigo, despite he has already lost to him. Badly injured, transformed back to his weaker form, he still tries to attack Ichigo. Until Nnoitra attacks him with his big freaking weapon. Grimmjow survives.
Nnoitra himself as well. Due to his nature as a Blood KnightDeath Seeker, he won't stop attacking his opponent until one of them dies. He's scared from Kenpachi several times during the battle, but it doesn't prevent him to attack his enemy. Just telling this Espada that the fight is over and walking away, after you just cut four of his six hands off (in the manga) and sliced him from his shoulder to his chest, just pushes Nnoitra's Berserk Button.
Baraggan, even when he is dying, still tries to kill Aizen and throws his axe at him.
Chad/Sado attacks and attacks Kyoraku, who is much faster and stronger than him. Even though, he never hits him, Kyoraku is very impressed by Sado's determination. Also, when Sado is badly cut by Nnoitra, he still continues his fight until he loses his consciousness.
Sui-Feng loses her left arm by Baraggan, but that doesn't make her surrender. She even uses her Bankai twice, despite her limit is once in three days.
Son Goku. Before the Cell Saga, he didn't know the meaning of the word surrender! And even when he did then, it was because he knew what he was doing. However, the greatest example was fighting Kid Buu at Super Saiyan 3, which consumes a hell of a lot of energy. As intense as the fight was, Goku was losing energy...and fast. He finally collapses from exhaustion, leaving Vegeta, to take his place in the fight instead. It was for five minutes, or less and Vegeta found himself nearly killed only for Goku to save him and transform into Super Saiyan 3 again, and battle him.
Another good example would be his fight against Piccolo - he's not going to let something simple like having a hole blasted through his shoulder and losing the use of all of his limbs prevent him from winning.
Vegeta is a bit of Determinator himself. Perhaps even more than Goku, part of why he's such a badass is the insane amount of punishment it takes to bring him down. Seriously, he gets his ARM SNAPPED IN HALF, but when Trunks tries to jump in and help he's all like "NO STAY BACK I GOT THIS."
His saga as a villain proves this beyond a shadow of a doubt. No matter what any of the z-fighters throw at him, he simply refuses to stay down. He gets knocked around by Goku's Kaio-ken, knocked into the sky with the Kamehameha— overwhelming his own attack— and the most powerful move in the anime, the Spirit Bomb. He had his tail cut off, reverting himself from his Oozaru form after expending the energy needed to create the power ball letting him turn into it. And then he fought Gohan in HIS Oozaru form and beat it. No matter what happened, he just kept going. Even when he was finally defeated, he didn't stop moving. He crawled to his spaceship to flee so he could heal up and fight another day.
Vegeta then one-ups in the next saga by wasting all his energy in an attack on Freeza, taking a subsequent beating from Freeza, and then after coming within an inch of his life, mocking Freeza after Goku's arrival, which prompts Freeza to blow a hole directly through his heart, and then wills himself to live long enough to explain to Goku about what Freeza did to their home planet, prompting Freeza to Lampshade the fact that he's still talking even though he's been blasted through the heart in both the dub and original, and making everyone realize that he isn't quite the monstrous bastard they originally thought he was after crying and begging Goku to finish off Freeza so that he will never turn anyone else into someone like himself.
Vegeta outdoes himself again in the final saga (twice), fighting Fat Buu by himself. Buu does not kill him. Instead it took Vegeta HIMSELF to kill him. He fights Kid Buu later on, this time only to buy time for Goku to "charge up." Cue the most brutal No-Holds-Barred Beatdown in the entire series. But Vegeta just kept getting back up, taunting Buu as he did so.
Son Gohan inherited the Determinator trait from his father.
Every last Z Senshi during the Kamehamehastruggle between Gohan and Cell, in the anime. By this point, even Piccolo is completely outclassed in terms of power, but he and the by-this-point-barely-relevant human warriors just keep blasting Cell from the sidelines, not caring what happens to them if they can do something to help Gohan.
"Why can't you people JUST STAY DOWN?!"
Freeza pretty much typifies this trope. Let's see, he engaged in a No-Holds-Barred Beatdown with Goku after he went & killed his best friend, prompting Goku to turn SSJ, and then he was in a losing battle with Goku, so he decided to blow up the planet they were on to win. At this point, he's out of energy & badly injured. So, he decides to play a deadly version of double-kienzan Frisbee with his opponent...which backfired spectacularly. On the verge of death from being vivisected, he finally breaks down & starts begging. Goku of course gives him his own energy out of spite mercy. Guess what he does with it? Promptly tries to shoot Goku in the back. This finally provokes Goku to blast him into oblivion...or so we're led to think. About a year later in canon, we not only find that he's managed to survive planet Namek EXPLODING, he got stronger, & he's come back for more. Understandably, everyone was quite shocked that he had survived, considering that about two thirds of his organic body was destroyed, even some parts of his head. The anime also shows that he was keep talking in space, despite being unconsciouness, before he was found by his father.
Tenshinhan deserves special mention for coming out of freaking nowhere with a long string of Kikohos - just to keep Cell on the ground temporarily. (The Kikoho saps huge amounts of energy, to the point that it's quite possible to die from exhaustion when using it - Tenshinhan keeps it up for several minutes before he passes out.)
Krillin has shades of this, too - the little guy gets flattened on a regular basis, but is just as regularly seen dragging his bleeding self off the ground to keep fighting, even when he knows that the best he can do is buy time.
Videl. When she wants to do something, she does not give up. Ever. Even when it's clear that she has no chance of accomplishing the task at hand. Her fight with Spopovich at the World Martial Arts Tournament is the best example of this one could give. After it became clear that she wasn't going to win, people kept telling her to give up. It took Spopovich's partner telling him to stop torturing her because they had more important things to be doing to finally end the match by him kicking her out of the ring. After a certain point, he was just torturing her and she wasn't moving on her own anymore, but, given the opportunity to stand up again, she would've surely continued attacking the guy.
Gloriously subverted with Android 19. He seems like this at first, but...
You'd better give up...THINKING I'M GOING TO GIVE UP!
In Naruto, trying to list all of his examples could fill a separate page:
The title character's "Way of the Ninja" basically consists of this trope.
Hinata is one of the first people to adopt Naruto's ninja way. Knowing that he's watching, she refuses to give up against Neji despite severe injuries, even standing up one last time after the referee ends the match to keep her from getting killed.
In the anime she got to do it again and even more impressively when she fought FREAKIN' PAIN! Unlike in the manga where she gets effectively One Hit Killed (though she actually survives), in the anime she keeps getting up... and getting up... and getting up...
Sakura, in her fight against Sasori, gets out of a cloud of poison gas by detonating a bomb against herself to blow away the gas. This entire battle is Determinator moment after Determinator moment for her, including a scene near the end where she stays standing and continues healing herself while a poisoned sword is completely impaling her, and even tries to insist that Chiyo use the last antidote on herself instead of helping her.
Just to remind you, you don't die on her watch. She will cut open your chest and give you a heart massage to keep you alive if she's out of chakra to do it the ninja way, dammit.
Rock Lee sometimes out-Narutos even Naruto, once assuming a fighting stance even though nearly all the major muscle groups in his body were torn and his left forearm and leg had been completely shattered. While unconscious, presumably from the pain. In his next battle, he breaks out of the hospital immediately after massive reconstructive surgery to saveNaruto from getting into a Hopeless Boss Fight with Kimimaro.
During his match against Gaara, his mentor was shown telling him in a flashback that his perseverance is what makes him a genius. This is echoed by Kakashi when he sees Lee pull off a Dangerous Forbidden Technique through sheer effort alone.
Taken to extremes in the Sasuke Retrieval Arc, where no less than five characters pull a Naruto. Even Brilliant, but Lazy Shikamaru breaks his own finger to dispel an illusion.
In the same arc, Kimimaro Just Won't Die. First, he somehow survives being crushed inside a ton-sized ball of sand (which splatters almost everyone it hits like a bug on a windshield) and keeps fighting. Then after being crushed a second time under a desert's worth of sand, he wills himself to life long enough to almost kill his opponents with one last attack. Oh, and he did all of this while suffering from a disease that was on the verge of killing him. Note that it was even stated that he was literally moving on willpower alone.
Jiraiya. He brings himself back to life by willpower alone long enough to give Naruto one last clue to defeating Pain.
Ironically, "GIVE UP TRYING TO MAKE ME GIVE UP!" was originally Nagato/Pain's line. And it probably inspired this line in the book Naruto got it from.
The Raikage. When confronted by Sasuke's incomplete Susano'o ultimate defense, coated by Amaterasu's unquenchable flames, the Raikage punches through it anyway, sacrificing his dominant arm in order to hit Sasuke. A later move would have done the same thing to his leg, but Gaara stops him from turning himself into The Black Knight. And losing the arm doesn't seem to have slowed him down a bit since then.
Danzo. Despite all these years, he never gave up on becoming Hokage, and he is utterly determined to kill Sasuke, even going as far as to use one last Taking You with Me attack on him (That didn't work.)
Orochimaru, due to his obsession with immortality. By the end of Sasuke's arc, wasn't he killed about three times? And he was significantly weaker than he used to be in his fight against Sasuke, who admits to that being the reason for his victory.
Sasuke's arc ended with Orochimaru killed in a definitive way by Itachi. Orochimaru returned from that. Granted, he needed help this time, but he still returned.
Ironically, the Big Bad Tobi/Obito has shown to be the biggest determinator in the series. Nothing, absolutely nothing will stop him. The destruction of his organization; the five great shinobi nations, the immortal zombie Hokages, All-Loving Hero and his own eye battling him at the same time and beating him to an inch of his life or even Madara betraying him and robbing him of his life support. He will simply get up and keep going.
Losing half of his body at age 13 only slowed him. So far he has been: struck with a fully powered Rasengan from the Fourth Hokage, poisoned with microscopic insects, set on eternal fire from Itachi, hit by 600 billion bombs from Konan blowing up at his face, pierced in the heart by a Raikiri enhanced kunai from Kakashi, taken over by hisEvil Mentor, Madara Uchiha, through the plant side and cut from shoulder to waist by the revived Fourth Hokage a second later. Once he becomes the Jubi's Jinchuuriki this is taken Up to Eleven as he takes a dozen mountain size gate seals from the First Hokage (a.k.a. God of Shinobi) dropped at his head at supersonic velocity before blowing them up with just raw strength. Not even the might of the Jubi ripping his mind apart gave him pause; he subdued it by the clinging to his love of Rin and the desire to recover his former team. Bonus points for doing with his own willpower what took Naruto help from his parents to subdue a weaker Bijuu.
Usopp; not only is he the weakest member of the Straw Hats (And according to Word of God, will always be the weakest, no matter what), he is completely normal. No Devil Fruit. No Haki. No Charles Atlas Super Power. Completely. Effing. Normal. He survived a 4-ton bat to the face, multiple explosions, having almost every bone in his body shattered, and got up to continue the fight, winning through cleverness and sheer determination. This after traveling through the desert for several days (admittedly, so had the others).
He is a Charles Atlas Super Power after his 2 years training for the new world, he's just still the weakest member of the crew.
No one who's travelled the Grand Line for this long can be considered normal, though. Usopp and Nami are at least Badass Normal from sheer experience alone.
As of the latest chapters, we now know why Whitebeard is known as the World's Strongest Man. Even though he's been stabbed and slashed 267 times, filled with over 560 bullets, hit by 46 cannonballs, and having half his God damn face melted off, he continued to mow down everything in his path, only stopping when he dies standing up.
And in testament to his pride, upon the back of his body, throughout his entire life as a pirate, there was not even a single wound telling of displays of cowardice.
Most people who carry the "Will of the D." would probably qualify as Determinators as well (or is it "D.terminators"?).
Luffy...just...LUFFY. Sure, he's the title character and thereby required to win in the end, but he's gotten his ass handed to him multiple times, and all of those times he's pulled himself out through sheer willpower. During his third fight with Crocodile, he got injected with a poison that could MELT SOLID ROCK, collapsed from it, and then got back up again after he should have been long dead. Not to mention, he took Lucci's most powerful attack at point blank range that damaged him INTERNALLY and even then, he refused to let himself even touch the ground!
Zeff "I've seen one or two like him before. Stubborn brats who'd rather die then turn their back on a fight once it starts...I'm glad he's on OUR side because people like that are hell to fight."
This trait of Luffy's is eventually deconstructedsohard in the Impel Down/Whitebeard War arc. His determination to save Ace is so great that he travels to Impel Down where he is curb stomped by the warden, goes through a procedure that shaves off some of his life span, recieves a temporary adrenaline superboost, has to claw his way out of the prison after missing Ace, then travels to Marineford, where an armada awaits. He has to fight through the Marines, including TheThreeAdmirals, the Seven Warlords of the Sea and his own grandfather. And despite all the obsticles he has to push through, He still didn't save Ace!
Well, technically he did, but Ace pulls a heroic sacrifice to save Luffy instead.
Zoro. Good God, Zoro. This man has experienced every form of bodily trauma just short of decapitation. You name it: slashed, stabbed, burned, exploded, crushed, bludgeoned, buzz-sawed, took in a pure distillation of pain. He is the poster-child of Overdrawn at the Blood Bank. At one point, when his leg is trapped during a fight, he decides cutting off his own leg to keep fighting is a perfectly rational thing to do. Nothing stops this man. He doesn't even have any powers, like HakiSpoiler UPDATE: Zoro can now use Haki after the Time Skip, so he's even more formidable now! or a Devil Fruit. He's just thattough.
Sakazuki is a great villanous example of this trope. After being handed a No-Holds-Barred Beatdown by the World's Strongest Man, he was presumed dead by many both in-universe and out. Cue shocked faces when he gets up, bleeding from the mouth, and starts wrecking shit with his Magma Man abilities again.
Sakazuki: "The minute I said 'I'm not letting them get away' ...it means you need to abandon the idea that he's going to survive, shit-for-brains."
Ah! My Goddess: Lind. Most goddesses faint due to the sheer feeling of loss and shock when their Angel is ripped out of them, but when confronted with the fact that the angel eater has latched onto her own angel, what does she do? Why, rip her own angel out. Granted that she had a second angel, but she gives that away to Keiichi to a last minute gambit to defeat said Angel Eater. And when later confronted with a possessed Belldandy during the same arc, she tells her allies (one of whom still had her remaining angel) to deal with the Angel Eater and leave demon-Bell to her, all the while knowing that the only way that she could do this was to use booster patches that would eventually leave her unable to fight ever again.
Airmaster: Sakamoto Julietta In one fight he gets his both legs and right arm broken, and yet he still stands up and attacks his opponent with his left arm... and then a kick. The referee decides that the match ends in a draw (as both are unable to move at the end) though some spectators disagreed as he has fallen a bit latter.
What was his motivation for all this? He wanted to go to his beloved girl.
Akagi pursues outrageous, life-threatening gambles with absolute determination. Part of it is due to completely trusting in his abilities (which would already qualify him for this trope), but his total disregard for the possible consequences of failures even when large amounts of luck are needed has forced every single character, including a batshit insane serial-killer, to call him "a devil" and "a lunatic with a deathwish".
Eren Yeager in Attack on Titan. In a bit of a deconstruction, his refusal to give up nearly gets him killed during his first Titan battle. In a bit of a reconstruction, that same refusal to give up activates his Titan shifter powers, saving his life.
Eiichirou in the sports mangaBaby Steps shows shades of this early on, as when he can't remember how he made his first "perfect shot" in tennis by accident, he practices for hours against the wall every day until he can duplicate it.
Bakemonogatari: Koyomi Araragi, helped in no small part by his vampirically-gained Healing Factor. Notably, whilst fighting the demonically-possessed Kanbaru, he ignores having several of his limbs broken, having his teeth knocked out, and having a hole punched through his abdomen. He only stops fighting- presumably due to being physically incapable of it- when he is swung around the room by his intestines.
In Bakuman。, Nobuhiro "Taro Kawaguchi" Mashiro never gave up, continually trying to make another successful series even after his contract expired. His brother, Moritaka's father, finds the idea that he could have been Driven to Suicide, a misconception Moritaka had for years, ridiculous, as he does not believe Nobuhiro was the type to give up.
Shujin and Saiko as well; they've perpetually been in Eiji's shadow for 8 years, and yet never given up on firmly taking the number-one spot.
Nizuma, too. Whenever it looks like Shujin and Saiko will FINALLY take the commanding lead, that's when he REALLY starts getting interested, and cranks the awesome meter Up to Eleven. The worse the situation, the more determined he is to retain his position.
It's even invoked right on the first volume when Moritaka talks to his dad on the phone: he reminds him of Ryoma Sakamoto of Star Of The Giants ("If a man should fall, he must fall forward, even if face-first into the mud") and Joe Yabuki (who protagonizes Moritaka's own favorite series).
Battle Angel Alita: Last Order: Niz holds "persistence" as his core tenet, and manages to impress the local Badass, Sechs, enough to learn what she can from his philosophy. Considering that Sechs had much more combat experience, was more physically capable, was kicking his ass for most of the fight, and on top of that was notorious for ignoring anything apart from a straight-on berserker assault, the mere fact that Niz made such an impression on her was pretty amazing. Then again, remaining standing and not crying or grimacing in pain after your arm, artificial heart, and half of your chest is blown away would probably have an impact on whoever you were fighting.
And let's not forget Gally/Alita herself. She's quite the dogged survivor and determined fighter -she has even survived war on Mars, being sentenced to dying in outer space two hundred years before the events of Gunnm, nearly getting mashed and thrown to the garbage on Chief Bigot's order, being blown apart by Desty Nova, and, depending on whether you're reading Last Order or the ending of the first Gunnm, fusing with the Ladder or being entirely destroyed at the nano-level while looking for the fata morgana. On top of all the more 'regular' fighting and danger, which include the destruction of several of her cyborg bodies. And let's not forget that * just* achieving the Panzer Kunst and the whole training around it is a feat in itself.
Berserk: Guts is a pure example of this trope. No matter how hellish his life is, no matter how many soldiers he faces, no matter how large and invincible the demons he fights are, no matter how seemingly impossible his goals are, he will simply not give up! It probably helps that he knows what will happen to him if he dies, and that he needs to complete his revenge against Griffith before he gets taken out.
There are numerous examples of that in the manga. Just to make two examples, Guts chisels off his own arm with a broken sword to free himself from a demon's jaws during the Eclipse where he is made a sacrifice. Later on, he gets both his legs and his arms (the remaining one and the artificial one) broken with ease by the God Hand, a group of 5 major demons that are actual gods of evil, and he, despite the obvious difference of strength that makes him look like an ant confronting an atomic bomb, rises back to his feet and uses HIS TEETH to swing his humongous sword, the Dragonslayer.
There is also a lot of emotional drive behind Guts' determination. Perhaps the saddest example came during the Eclipse, when Guts sees Casca, his lover, about to get the, ah, tentacle treatment by a mess of demons, starts to go into a Foe-Tossing Charge to get to her, gets his arm nommed by a demon with a set of impressive choppers, then hacks off that arm we were talking about when Femto starts raping her, and then continues his rampage to try and save her... But it was all in vain. But even after he gets dogpiled, Guts is still trying to get to herby trying to force himself up from beneath the demons on his bloody stump of an arm which evidently causes one of the demon's talons to claw out his eye. After all of that occurred, Guts not only remains a determinator to ensure his own survival, but also Casca's survival, since if he dies, she also dies.
Guts was a determinator even as a baby! Guts was born after his mother was hung from a tree, and despite being exposed to the elements for who knows how long before a band of mercenaries found him, he absolutely refused to die. He's been fighting for survival his entire life.
Black Lagoon: Roberta. Terminator jokes are made, but saying them will be regretted to a point where any Shounen protagonist you want elected for this trope should just step aside.
Blame!: Killy. Broken arm? Use the other one. Three-fourths of your body incinerated? Grow a new one. Half of your skull blown off? Just shoot back. Seriously, there is nothing that can stop this killing machine. Now add in the fact that his quest takes 3,000 years to complete, during which he does not complain or quit once. Furthermore, he walks through the majority of The City. The City is roughly a light-year across.
Busou Renkin not only justifies this behavior in its protagonist Kazuki, it uses it as a major plot point. The kakugane serving as Kazuki's heart is a highly dangerous, modified version, which has given Kazuki energy-draining abilities and is slowly turning him into a monster.
The protagonist of Chirin No Suzu is a lamb whose mother dies early in the film. He won't stop at anything to be powerful like a wolf. He even grows up into a murderer.
Chrono Crusade: Rosette Christopher, in the manga. She'll do anything to get her brother back, going as far as making a Deal with the Devil that would ultimately cost her much of her life to do so. She even comes Back from the Dead, almost entirely by sheer force of will.
Claymore: Clare is a positively epic Determinator. She can get filled with holes, lose her limbs, knocked down over and over again, and it just.Doesn't.Matter. While considered a Gecko Ending, her climactic battle with Priscilla in the anime is the best example of this.
Though, her fight with Priscilla in the manga might be an even better example. When Clare is assumed to have been killed by the Destroyer, she ends up taking control of it and attacking Priscilla. There's a reason quite a few people consider her the Distaff Counterpart of Guts.
Code Geass: Kallen Kouzuki and Jeremiah Gottwald are competing fiercely for this title, with the former having succeeded in fighting Suzaku to a Combat Breakdown standstill in the Grand Finale, and the latter having survived everything from Kallen blowing his mech to hell to getting hit point blank with an EMP device specifically designed to work on the mineral his later cyborg parts were powered by, and continuing to keep moving by sheer force of will. And what's left of his human muscles, of course.
This makes it fitting that the first R2 opening shows Jeremiah while screaming: "I CONTINUE TO FIGHT/FLY!! I CONTINUE TO FIGHT/FLY!!!!"
Crest of the Stars: Almost every Abh embodies this trope, but for Captain Lexshue it becomes a Crowning Moment Of Awesome. When her ship discovers a fleet of warships trying to launch a surprise attack, she orders Lafiel to escape with Jinto and warn their allies. Then to buy them time, she turns and fights despite being outnumbered ten to one, and keeps fighting even as the ship is being destroyed sector-by-sector and finally crippled. As she's ordering her beloved crew to ram the last ship, a beam pierces the hull in slow motion and the scene fades to white.
Both Light Yagami and L of Death Note are known to be "childish, and hate to lose" and "don't know any limits." Both are convinced that they are in the right and will stop at nothing to prove that their "justice" is correct.
D.Gray-Man: Allen and Lenalee in particular push their innocence past all reasonable limits, willing to sacrifice themselves in the process to save their friends. Allen also has his promise to Mana to never stop walking the path he's chosen until he dies.
The Fourteenth is arguably this too seeing how he devised a way to come back to life just so he could kill the Earl with his "own" hands.
Digimon Adventure 02: Daisuke/Davis Motomiya - "No! I'll never retreat! Not as long as there's a fighting chance! I've never quit before, and I'm not going to start now!"
Ditto Masaru from Digimon Savers for being the first human in the history of the franchise to punch a digimon and do real damage. He does this several times, in fact he broke through a shield made by the god of the digital world (even if it had been weakened) with a punch using his determination
Impmon of Digimon Tamers survives literally getting curb-stomped and thrown in lava through sheer stubborness, and is incredibly durable as Beelzemon. He also digivolved into Beelzemon Blast Mode with just his willpower—no Digimodification, no Calumon, just pure determination to help his friends.
Played straight by several characters in Elfen Lied. Nana in particular is on the receiving end of several Curb Stomp Battles, but still gets up for the next one. Bandou doesn't let the loss of his eyes stop him from being one of the biggest BadAsses in the story. Lucy is this and especially in the manga finale. The Agent demonstrates this in every fight she's in looking for ways to survive and win even in the final scene she's in, trapped in a cave with the Big Bad's daughter, after a brief discussion, she turns to a hulking tumorous mass, and asks if its edible. That last scene Makes Sense In Context.
In Eureka Seven AO, the Secrets are hellbent on chasing the Scab Coral to whatever timeline and dimension they went in order to eliminate them.
Eyeshield 21: Subverted. In one early match against the Oujou White Knights, Hiruma stops playing because he realizes Deimon's chances for victory have reached 0%, although he is willing to keep playing in any other match as long as his team's chances for winning are even marginally greater than nil. In recent chapters, he's gone from being a subversion to almost an embodiment of sheer determinatorism: He plays half a game with a broken arm and apparently heals it in less than a month by willpower alone. Well, willpower and his oxygen tank/segway. Hahahaha.
Then there's Yukimitsu Manabu, who has more determination than any of the other Devil Bats, and managed to overcome Agon in the game with the Shinryuuji Nagas. Ojou's Sakuraba and Takami both elevated themselves into great players by never giving up; Takami, particularly, had to overcome a lack of speed that many said killed his chances of being a quarterback. He compensated by practicing the living Hell out of throwing until there wasn't a more precise QB in the country. Monta managed to overcome the Nagas' catching ace, as well. Because catching is all he's good at, he's vowed to become the number one catcher in the world if it kills him. Interestingly, characters who started out hearing threats of "do it, or I'll kill you!" (Sena, Monta, the Ha Ha brothers, etc.), end up deciding for themselves "I'll do it even if it kills me!"
In Fairy Tail, pretty much everybody from the eponymous guild is a Determinator. Natsu and Erza in particular stand out for taking the determinator status to ludicrous levels.
Kariya Matou from Fate/Zero really tried so damn much to save the daughter of the woman he's loved from Fate Worse than Death. To that end, he willingly endured Zouken's parasitic worms for an entire year even though he had never been educated in magic.
"Sometimes, a broken machine did not simply and silently give up functioning. On rare occasions, it could surprisingly continue working. The fact that Kariya was able to crawl back to the Matō mansion in Miyama was one of these very rare examples."
Fist of the North Star: Juuza and Fudo definitely exhibit this in their respective fights against Raoh. Despite all the punishment they're put through, they absolutely refuse to surrender. At points in both fights, Raoh specifically states they should have already died from their injuries, yet they continue to fight on, even in death.
Raoh, Souther and especially Kenshiro himself also qualify big-time.
Also, Rei. He had to kill Yuda, but due his earlier fight with Raoh he was dying in horrible pain. And he soldiered on. But Yuda was still running, so he had Toki treat him in a way that could extend his life for a while... Assuming he survived the horrible pain, ten times worse than what he had experienced from Raoh's attack and concentrated in a few minutes. He survived (even if his hair went white from the pain), and continued fighting, living until he had killed Yuda (and doing so in spite of Yuda flooding the town to rearrange the battlefield).
Freezing: Satellizer L. Bridget is a true female pinnacle of this trope. Because of her late mother's dying wish, she refuses to give up in any situation and refuses to be bossed around by anyone. Even with her body paralyzed or brutally maimed, or even after having her throat slit, she will not give up. She has pushed her will so high that she defeated a superior opponent while unconscious (she has "dead eyes" during the latter stage of the fight).
Edward. OH SO MUCH. Not one injury he receives during the entire series is enough to make him give up on a fight. Ever. Special mention for his final fight with Pride. The one where he invaded Pride's philosopher's stone to make him stop trying to possess him, while Pride was pulling a Grand Theft Me via shadow tentacles burrowing into his skin.
Mustang has to endure the loss of his best friend, the crippling of one of his best men, having the rest of his most trusted men reassigned all across the country by Fuhrer Bradley as punishment for sticking his nose where it doesn't belong, and having his eyesight forcibly removed by Truth for a human transformation he didn't initiate. And he still keeps on trucking right through the final battle.
Lan Fan also deserves special mention. She is one of those people willing to cut her own arm off for the cause, and does not give up on the Young Master even after he is turned into a homunculus. She went through automail rehabilitation (this normally takes upwards of two to three years) in six months, cutting Ed's record of one year in half.
Let's not forget Kimbley. They guy got his soul added to Pride's Philosopher Stone and is essentially dead. Then, when Ed is in the final fight with Pride, Kimbley's soul emerges from the legions of other souls to stop Pride because he had lost his fierce pride by trying to take over Ed and thus lost the fight in Kimbley´s eyes.
Sousuke of Full Metal Panic! is a prime example. He’s done everything in his power to overcome his very difficult life in the name of sheer survival. Let’s account the events of the final arc alone: when Leonard kidnaps Kaname, Sousuke embarks on a long journey to bring her back. He loses all backup and watches friends die. He is hunted by a multitude of assassins and almost dies several times. He is revived just so he can fight off more assassins. He endures rigorous training, prompting the little gem below. He storms enemy bases and loses Kaname again and again. He’s betrayed by a deeply-trusted comrade. The guy doesn’t even give up when the girl he does all this for seems to have joined the enemy to work against him. He’s not giving up, whether the object of his search (and everyone else in the world) likes it or not.
Lieutenant Colonel Courtney: "He's got what it takes. Just fucking guts."
Courtney: "No, fucking guts."
Some of the racers in Future GPX Cyber Formula are this, but the biggest one of all is the protagonist, Hayato. He has a "hate-to-give-up" mentality that gets him out in most situations.
Shinjyo also has his moments. In episode 32, near the end of the African GP, his Fire Superion has stopped running when it runs out of gas as the car is about to cross the finish line. So he gets out and he pushes his car into the finish line. He learned this trick from his F3 days, when his car also stopped.
Gantz: Katou Masaru is a determinator All-Loving Hero, of the "Leave no man behind!" and "There is no 'Combat Ineffective' in 'Sudden Battlefield Limb Amputation'!" variety. In one case he beats an unstoppable alien demigod equipping his own severed arm, and in another he crawls over to a BFG, props himself up on his bleeding leg-stumps, and cuts loose on yet another unstoppable alien demigod. His favored tactic? Talking people down. And he's not even the main character. Main character Kei Kuruno isn't far behind, except for the All-Loving Hero thing at the beginning.
GaoGaiGar: The Gutsy Geoid Guard actually had written into their bylaws (Called the "Oath Sworn through Courage") : "Article 5, Section 120: ...Under no circumstances must a GGG member ever give up."
ChoRyuJin = Determinator Incarnate, even out-"never give up"-ing Guy. Even THE END OF THE (prehistoric) WORLD cannot and will not stop him.
But even ChoRyuJin faltered before the might of the Sol Masters in FINAL. In the darkest hour of the final battle, with the Brave Robot Corps in pieces and Guy seemingly defeated by Palparepa, it was Mamoru, alone and armed only with his purification abilities, who refused to surrender in the face of Pei La Cain's superior firepower. It was Mamoru's unflagging courage that empowered the G-Stones of the Brave Robot Corps via the "all G-Stones are connected" reveal moments later, giving 3G's warriors the strength to come back from defeat one last time and blast the Sol Masters to kingdom come.
Gintama: Pretty much all characters in main cast have their moments, but the all-out Determinator has to be Gintoki. When things get serious, he shows this more focused side of his and it's often combined with few Crowning Moment Of Awesome moments. Not very surprising, since he is the protagonist of shounen series.
Great Teacher Onizuka embodies this trope. The best example is when Onizuka has to substain a student national examination to avoid forced resignation, and he manages to rescue a girl in danger from a dozen yakuzas, armed with guns, just with bare fists, then he arrives to the 5-hours exam 4 hours late and he does it in 1 hour. After completing it, he passes out and the other characters realize that he took the exam WITH 4 BULLETS IN HIS STOMACH.
Domon Kasshu of the Mobile Fighter G Gundam is probably the best example, but in G your fighting spirit is your most powerful weapon anyway. Sai Saici deserves a special mention for continuing to fight after his Gundam's arms were ripped off. Keep in mind that Gundams in this show work through transmitting the pilot's movements directly to the machine—so Sai Saici actually felt like his own arms were severed.
Gundam 00's Determinator surely is... Patrick 'the Immortal' Colasaur unwilling to stop fighting Gundams (and for his love interest) no matter how often he gets shot down. Though, there is Graham Acre (aka Mr Bushido) who keeps fighting up to the point where he 'overcomes reasoning with his recklessness' - and it works. Also, Setsuna. Most people who experienced his childhood would have suffered from massive PTSD for the rest of their lives, but he is able to break away from his past and start looking towards the future through sheer determination. This trope is also the leading cause of Setsuna Taking Many Levels in Badass throughout the series, not least of which is when he activates the Trans-Am Burst on 00 Raiser while screaming "Everyone's lives are fading away...I...am NOT GONNA LET THAT HAPPEN!"
Garrod Ran from Gundam X is another example. This trope is the entire reason why Garrod is able to debunk Newtype prophecies again and again. D.O.M.E. even says this outright in the final episode.
Heero Yuy of Gundam Wing. Shoot him, blow him up, he's gonna complete his mission even if it kills him. Though that's pretty unlikely, even he can't manage to kill himself. He screams "I WILL SURVIVE!" as Wing Gundam Zero literally dissolves around him, re-entering the Earth's atmosphere unshielded and backwards, in the series finale. Turned Up to Eleven in Super Robot Wars W where the only way to fire the two cannons at the same time is by that method.
Guyver: Sho Fukamachi, unlike many of these heroes listed, does not seem to be very strong willed, that is until his friends and family are threatened. He frequently suffers from some of the most horrible injuries, including limbs being torn off, getting impaled, his skull smashed open, and even melting and dying (thankfully the Guyver can regenerate), but he always keeps fighting. In his rematch with Aptom, he manages to get over the trauma of killing his father to bio-boost again and save Mizuki, and he wills his megasmasher to open and fire when his arms are disabled. Much later, Sho fights for control of the Gigantic from Agito, The Ace, and wins through his sheer will power. This is notable because Agito himself is also quite the determinator who will do anything to win.
Gyakkyou Nine: Fukutsu Toshi's name literally translates to "Indomitable Determination." Being a Kazuhiko Shimamoto character, he is a walking inferno of hot blood and believes that with determination, his baseball team can achieve anything.
Hajime No Ippo: Virtually every boxer Ippo faces shows a will of steel and Heroic Resolve. The "determination" is even commonly seen as a green flame in the eyes of the person. However, the title of Determinator must go to Takeshi Sendo who, in one fight, after fracturing several ribs, not only fights savagely for a whole 'nother round, but after being brutally KO'd (and shattering two ribs in the process) he stands up, simply refuses to let his body go unconcious just so he could stare directly into Ippo's face and later walks out unassisted.
And Date Eiji, who kept fighting with a broken rib, fist and jaw against the world champion.
An interesting variant is Ichirou Miyata, who combines a will of steel with low stamina and durability. He really can't take much and his weight management makes his stamina even worse, but he just won't give up regardless.
Hellsing: Alucard suggests that this is part of the reason why Humanity fascinates him, after Father Alexander Anderson refuses to roll over and give in despite having a partly blown off left arm and being severely outnumbered.
A more awesome example is provided by Pip Bernadotte of the Wild Geese:
Pip, while wounded in the stomach, cracks a Vampire Nazi over the head with a rifle butt, shoots them (ineffecively) with a pistol several times, then fires smoke grenades for cover, picks up Seras and carries her down the hall while having his legs shot at, whilst he's losing blood from the wound in his stomach, and doesn't stop until Zorin Blitz throws her scythe at him and impales him on it.
To its credit, the anime (as opposed to the OVA) depicts a flashback of Integra, aged around 12 or 13, facing down a newly rejuvenated Alucard. Even after he instantly heals the wounds she inflicts with her pistol (and being absolutely horrified of him up to this point, not to mention he's currently right in her face) suddenly CUTS ALUCARD OFF in the middle of his "I am Immune to Bullets" speech, and delivers a kickass speech almost crying Berserker Tears. Amused by this sudden display of spine, Alucard formally recognizes her as her master (and thus the rightful head of the Hellsing Organization).
Yuu from Holyland. He's a Cowardly Lion through and through, but when the Lion is awake he will force himself to get up from things that are previously shown taking down lesser men and keep going until either him or the opponent is thoroughly down for the count. Things like getting thrown to the concrete by a judoka or being a Groin Attack victim.
Honey Crush: Mitsu doesn't even let a pesky thing like death stop her from pursuing her love interest. Her unwillingness to ever give up is lampshaded by her rival, Kyouko.
Hunter × Hunter: Gon makes everyone else look like pansies, mostly because unlike the rest of 'em, he's far from the strongest fish in the pond. This is a trait all Hunters share, since the final exam is to eliminate the least Determinator among them (it's an inverse fighting tournament where the loser moves forward, and the only way you can lose is by admitting defeat—killing an opponent will disqualify you). During this exam, Gon's opponent Hanzo clearly outmatches him, beating him unconscious several times, and breaking his arm. Even after hours of this, he still refuses to surrender. He finally wins as Hanzo realizes that Gon has won him over and that he's not willing to do what it would take to win (actual torture and permanent damage). As soon as he recovers, Gon pulls the same trick again when he's trying to find Killua—each time he crosses the boundary, he's smacked several yards back, but he keeps it going all day long until the person who's smacking him cries out in desperation, eventually persuaded to let him pass. A later arc has him getting his arm blown off—and counting that as a victory because he made his opponent use his special ability. Frankly, Gon's determination (or stubbornness) would probably get any non-protagonist killed within minutes. In the manga, he has gotten even worse since the death of Kaito
InuYasha: The entire InuYasha-tachi team. InuYasha himself, Sango, Miroku, Kirara/Kilala, even gentle Kagome and Shippo, who usually plays comic relief. The only time any of them could be said to give up is when Sango thinks Miroku is about to die from his curse. "Take me with you."
Kagura, who persisted through the most relentless circumstances, and didn't even let dying bring her down.
JoJo's Bizarre Adventure: This is half the point, for both flavors of 'tagonist. (The other half is being wicked clever bastards.) JJBA fights are traditionally won by the protagonist being put under an ungodly amount of pressure causing him to figure out something completely, well, bizarre and thus turn the tables on the unfortunate villain involved. Or sometimes to just stop time.
Deconstructed when Caesar fought Wham. After the latter managed to give the former an ungodly beatdown, he turned away, before catching the victim out of the corner of his eye, back on his feet and ready for more. Unfortunately, Caesar had lost far too much blood and wasted far too much energy that he wawsn't even able to throw any kind of punch with any kind of force behind it, let alone produce a Ripple. So, even though he refused to go down, in the end, standing back up didn't do anything at all.
This is even included as a gameplay element in one of the games. Once their health has dropped below a certain threshold, certain characters (Giorno, Bucciarati, Mista, Fugo and Ermes) enter "Resolve Mode", which gives them a bunch of advantages. They gain super armor when attacking, they do more damage, and they gain more energy from being damaged.
Jubei-chan: Koinosuke survives 300 years by force of will alone to keep searching until he finds the true successor for the Lovely Eyepatch. (He also abandons his wife and daughter to do so, but you don't find that out until later.) It's nearly the last Despair Event Horizon for Jubei when she realizes what he went through to get her something she's done nothing but reject.
Karas: Otoha definitely falls into this category. He was "killed" twice in the series that we saw—and both times just pissed him off. And yes, he was in human form both of those times.
The second time was stated that Otoha was revived by "the will of the city." It seems whatever entity governs the Karas was so impressed by Otoha's sheer dragging-on-the-ground balls that they refused to give up such a valuable guardian.
Yukino Miyazawa the protagonist of Kare Kano. She is an Attention Whore who wants above all to be praised by others. What makes her a determinator is the lenghts she goes in order to do that, always trying to be the most helpful, beautiful and the best in school and sports.
Katekyo Hitman Reborn!: Hibari Kyouya. Even when half of bones in his body are broken by Mukuro, he continues fighting so hard that he was said to be moving too fast to be seen. And don't forget about the time he was been poisoned by the Cervello during the Varia arc and is supposed to take the Cloud Ring that was held up above him twenty feet high. The poison is said to stop a rampaging elephant into a standstill, but not for Hibari!
Kenichi: His masters are firmly of the opinion that sheer stubbornness is the only reason he survives at all.
The same series gives us Akira Hongo, one of the members of the One Shadow Nine Fists group, Sakaki Shio's Rival Turned Evil and a Jerk with a Heart of Goldand Standards. Sakaki reveals via flashback that as a young man, and prior to his Face-Heel Turn, Hongo once got shot with a semi-automatic gun and sustained injuries serious enough to warrant a trip to the emergency room; yet he got up and left the hospital on his own power, went to a karate tournament and won even with the severity of his injuries—all because the day before, he declared that he was NOT going to go back on his appointment with said tournament no matter what. That same determination translates in the modern day to him being Impaled with Extreme Prejudice several times in his fight with fellow Nine Fists member Silcardo Junazard and yet not quitting until he himself stabs Junazard through the torso with his injured arm. "I'll be the one to kill you," indeed.
In Kinnikuman: almost all of the character's qualify. Buffaloman fought Akuma Shogun (The god of evil choujin) for a FUCKING MONTH while Kinnikuman (A determinater in his own right) trained to fight Akuma Shogun. Terryman fought Ashuraman after having literally lost both of his arms, he didn't win, but he still fought Ashuraman hard. He also went up against The Mountain (The heaviest choujin in the series by that point), who not only body splashed him, but also damaged Terryman's already injured leg. Terry come back and beats the Mountain by lifting him over his head and giving him a brainbuster down a casm they were fighting over. When Kinnikuman and Brocken Jr. couldn't find Terryman, they though him dead. Lo and behold, Terry survived the staggering fall. Near the end in the Scramble for the Throne Arc, Robin has to face Mammothman in death match. After Mammothman freezes Robin in an iceberg, Robin bursts out, steals Mamothman's own move, the procedes to give the Rope-work Special, all the while Robin is fading out of existence. Geronimo probably deserves this more title the most. Way back in his first fight with Sunshine, Geronimo's heart stops. Given that he's a regular human being, you'd think that would kill him, but hell no! Geronimo shoves his hand into his own chest and FUCKING MASSAGES HIS OWN HEART UNTIL IT STARTS BEATING AGAIN!
The Law of Ueki: Ueki just will not go down until he succeeds, no manner how much damage he takes. His absolute determination is so great that many characters have incredulously asked: "Is this guy immortal?" It helps that he is a cosmic being, albeit still killable, who was raised as a human, but even other cosmic beings are awed at his refusal to go down.
Lone Wolf and Cub: Ogami Itto. Enough said. Well no, not really. Er... go read the page!
We have Zenigata. Whenever he's chasing Lupin, nothing will stop him. Jailing him? He'll break out to continue the chase. Reassigning him to another case? He'll solve it while still chasing Lupin. Beating the crap out of him? He'll beat you up and continue chasing. A war is closing the border between the country he's in and the one Lupin is going into? We never saw what he did, but he was later seen leading the police of the second country while chasing Lupin.
More subtly, we have Lupin III himself: He always gets what he's after. No matter the challenges he has to overcome, they will be left in ruins by the time he's walking away with the treasure.
The titular character of Magical Girl Lyrical Nanoha is one hell of a determinator. She doesn't give up, even in the face of a Person of Mass Destruction that is clearly more powerful than her. Quite possibly deconstructed in that her constant pushing beyond limits bites her back hard at one point. She later teaches her students not to go down that path.
Exemplified in the beginning of the second season. She fights a stronger opponent, gets knocked through a wall and nearly killed outright, is told by her True Companions to stay put, heal up and let them handle the rest, and instead pulls out the biggest spell in her arsenal to end the fight - and manages to cast it despite an enemy mage shoving a teleporting hand through her chest to try and neutralize her.
Nanoha's intelligent device Raising Heart also appears to show shades of her master's Determinator traits, especially in the second season when she INSISTED that Nanoha cast Starlight Breaker despite the possibility that she may break completely and during the battle between the Person of Mass Destruction, INSISTED that they use her new form, again despite Nanoha's protests of her possibly breaking.
As mentioned above, at one point in her career, Nanoha suffered near fatal injuries, as a result of overexerting herself. Doctors (not earth doctors but Mid_Childian ones) told her that her injuries may lead to the end of her career and a high possibility of losing her ability to walk ever again. Yet she went on to become one of the TSAB's most powerful mages, earning the title of "Ace of Aces." That's right, Nanoha simply refused to go down.
Vita is another example. She is quite possibly an even bigger determinator than Nanoha, or at least it looks more impressive since she has insane endurance. Fighting through hordes of combat machines, getting stabbed right through the chest, fighting through more hordes of combat machines until she arrives at her goal, only to be greeted by automatic defense systems (laser shooting cubes) after her first attack fails. And her only response? "Bring it on.".
Magic Knight Rayearth: "Cephiro, the world that is shaped by the strength of the believing heart." And no one, absolutely no one, has a stronger heart than little pint-sized Hikaru Shidou, even when she's torn apart with injuries that would have killed a grown man. It's actually a plot point in the anime's second season, when her loss of will is downright catastrophic for the entire world, and her subsequent recovery makes her virtually omnipotent.
Parodied when Negi is accidentally turned into one of these after Nodoka asks him for a kiss, unaware that the nearby The World Tree forces him to accept. He's going to kiss somebody, dang it, or die trying. He eventually kisses Asuna so intensely that she does an incredibly suggestive Eye Take just before the scene is jokingly censored.
Right after Negi and crew arrive in the magic world, Fate attacks Negi, putting a stone spear through his chest. When he moves to attack the students, Negi pulls the spear out and beats Fate in the head with it.
Negi gets attacked by a shadow mage. During the course of the fight his right arm gets cut off. His response? "I still have my left!", before attempting a finishing blow. Unfortunately, Jack Rakan stops the fight before we find out if it would have worked or not. And don't worry, he gets the arm back.
The finals between Negi and Rakan. Good Lord. Negi's got Tears of Blood, a gaping chest wound and is puking up blood... and he still gets up to fight on. Even Rakan is surprised by his tenacity.
Negi's tendencies along this line showed as early as vol. 7. When he first asked Evangeline to train him she refused unless he could use the kung fu he started learning from Ku Fei a couple of days earlier to land a hit on expert fighter Chachamaru. At the appointed hour he came, got sent sprawling halfway across the courtyard, and was told to go away. He simply picked himself up and pointed out to Eva that no time limit was set. The resulting beating lasted for over an hour before Chachamaru got distracted and Negi landed a feeble blow before collapsing.
Then there's Rakan, who takes this to absurd levels. Nothing short of erasing him from existance will stop him, and even then, he puts up one hell of a fight. Even after that, he willed himself back just to give his student one important information that he must absolutly must know, and also gives some parting words to everyone
Haruka in the anime incarnations of Mai-HiME and Mai-Otome is the resident determinator. Whether kicking a tank as a normal human, head-butting someone while literally dissolving into green sparkles or blowing the head off a monster after getting hit by its petrification ray, Haruka does not know the meaning of the word "quit".
Mazinger Z: Kouji never gives up. Not even when he is going against several Kikaiju at once. Not even when the Robeast is completely impervious to his Humongous Mecha's attacks. Not even when it has abilities his robot could not match (he faced submarine and flying Kikaiju long before Mazinger got upgraded to be able to fly or swim or got weapons worked under water or in air). Not even when he is buried under rubble. Not even when he gets dumped into a freaking volcano! Not matter the odds, he will think fast, analyze his foe (and its weapons and capabilities), come up with a plan or cheat like crazy.
Geez, not even when he KNOWS the fight is utterly hopeless and there is no way to win he will quit. In the last episode of Mazinger Z and in the movie Mazinger Z tai Ankoku Daishogun he deployed Mazinger Z as stating he was not walking alive out of that battle and he knew that... but he did not care.
The only way you can get him stopping to fight is holding someone hostage. And even then he will try to exploit any edge to release the hostage and giving you a sound trashing.
Great Mazinger: Tetsuya is even greater of a The Determinator than Kouji. Not matter how hurt he is or how badly damaged Great Mazinger is, he will NEVER give up (because in his mind giving up would mean he would let his surrogate father down).
In the Great Mazinger tai Getter movie, the monsterGilgilgan ate Great Mazinger's left fist, shrugged a Breast Burn attack off, reflected a Thunder Break, blew Great Mazinger's right fist up and endured a Navel Missile, melted Great Mazinger's right leg... What did Tetsuya do? He tried to headbutt it. He tried his last weapon -Great Typhoon- and although it had no effect either, he YET refused leaving.
Tetsuya: That's right... It's fine if you tell me... to fight... I'll keep answering with the same answer... MAZIN GO...
UFO Robo Grendizer: Even though Duke is calmer than his predeccesors he is not a quitter, either.
In the Go Nagai manga he fought Mazinger and Great Mazinger at once in spite of all told it was crazy and hopeless, and he held his ground and even sliced one Great Mazinger's arm off.
And during the first season's two-part finale, Grendizer had been beaten by a pair of giant robots, Duke's nemesis Blacki had taken control of the Science and Space Center and was torturing Dr. Umon, Koji's flying saucer was too damaged to help and Grendizer was forced into hiding. Still, Duke did not give up. When he made his comeback, he engaged in a duel with one of the enemy mechs in a crumbling cave, he leapt up, shot lasers to bore his way back to the surface, decapitated the five-headed mech, and then flied straight at Blacki's command ship in a game of chicken. And he won!
Akise Aru of Mirai Nikki. In Chapter 51, he is stabbed and slashed in the neck by Yuno. Knowing that he's pretty much screwed, he decides to give Yukiteru a final warning about Yuno by texting it into his cell, running up a hill, and shoving the message in his face. And so he does. AFTER BEING DECAPITATED. What a Determinator...
Monster: Inspector Lunge obsessively chases Tenma all over Germany trying to get him to confess to the murders that were actually committed by Johan. Nothing - not even his wife and kids leaving him, or losing his job - matters as much as catching Tenma. In a more traditional example, not even being stabbed so badly that he quickly falls unconscious from the blood loss stopped Lunge from trying to smash the windows of Tenma's car in order to catch him.
Muteki Kanban Musume is a Genre Deconstruction of the Fighting Series, and Determinator is also deconstructed with Nishiyama Kankuro, the Unknown Rival who constantly defies Miki to a duel because she bullied him when Kankuro was a teenager (and Mike a child four years younger than him). Kankuro's hope of defeating Miki in a fair fight exists only in his head, as his refusal to give up is fairly foolish. In fact, he can easily defeat Miki working as a Poster Boy for the Ramen Restaurant, but he is so obsessed with the fair play that he never realizes it.
Neon Genesis Evangelion: Rei has her determinator moments, most notably in the climactic fight of Rebuild 01 where she blocks a particle accelerator beam that melts half a mountain behind her.
Somewhat subverted in that the reason for Rei's seemingly extraordinary courage and willpower is her complete and utter disregard for her own safety and wellbeing, as she can be replaced.
Surprisingly in Rebuild of Evangelion 2.0, Shinji. The moment when you see his eyes glowing red and you hear "GIVE AYANAMI BACK!!!" you know that not only has the story gone Off the Rails, but also that even starting the Third Impact won't stop Shinji from saving Rei.
Hell, he didn't even care that he would be reduced to LCL. He wanted her BACK.
Asuka is another Determinator downright refuses quitting. She shot a Macross Missile Massacre on Zeruel—that it shrugged off—, and when it cut her mecha's arms off, she tried lunging at it. She stated she preffered dying rather returning in defeat when Arael was mindraping her. She fought the nine Mass-Production models and she nearly won. And when they plucked her right eye out, pinned her on the ground with their spears and her Humongous Mecha was out of power she still was trying goading her EVA into a Berserker rage. She tried to battle until the very end.
Noein: Karasu is the series' Determinator. If you threaten Haruka, he will smack you down. It is that simple. In the last episodes he took this to the extreme, by taking on an entire universe just to get to her.
Now and Then, Here and There: Shu keeps repeating "Everything's OK!" in the most extreme circumstances, to the point that he might seem completely unconscious or heartless - as when he tells Sara that "everything's okay" because she's alive, even though she was spirited away from her world, unfairly imprisoned, raped by a soldier whom she had to kill, and ran away from a military fortress to avoid getting executed. Oh, and the rape resulted in a pregnancy that she tried to terminate by bashing herself in the stomach with a rock, until the ever-determined and optimistic Shu talked her into carrying it to term..
Orguss02: Verifer Decimator shrugs off absolutely everything the Rivellian army throws at it, from artillery to bombs aimed directly at its head. Even when it's finally damaged (by having nuclear weapons detonated right against its body) it only loses both legs, continuing to drag itself forward with its hands until it's destroyed by an even bigger Decimator.
A good example of how Ash absolutely refuses to give up, no matter how outclassed he knows he is, comes in the first movie. What does he do when confronted with Mewtwo disabling all the other Pokémon's abilities and forcing them battle clones of themselves, eliminating all the normal attack options? He charges Mewtwo and throws a punch at his face. When he hits a psychic barrier instead and gets knocked on his ass? Ash gets up and does it again.
The Team Rocket trio. Hell, they friggin' follow Ash EVERYWHERE, from the Kanto region to Sinnoh and appear in EVERY SINGLE EPISODE except the first, 99% of the time ending in them getting their asses handed to them and flying off into space AND THEY STILL COME BACK FOR MORE.
However, this waters down a bit in Best Wishes, where the trio leaves Ash alone in favor of getting a special mission from their boss. They still come around to bother Ash from time to time when Giovanni leaves them to their own devices, though.
Some Pokémon could also qualify, such as Scraggy, and to a lesser extent, Tepig. More "classic" examples include Squirtle and Charizard (though the latter only applies after he stops being a Jerk Ass).
Pokémon Special gives us Gold. Kick his ass multiple times? Doesn't matter. He'll still try to fight you.
Pearl is one too. It helps that he's closely associated with Azelf, the Pokemon of Willpower.
Ashitaka from Princess Mononoke fits this trope to a T. Remember— he was shot through the chest and yet still had the strength to open a gate that it normally takes 10 men to open.
Puella Magi Madoka Magica: Subverted with Homura, who tries to be one to save Madoka and fails spectacularly — turns out her determination didn't accomplish anything except bringing about the apocalypse. But then reconstructed when Madoka uses her wish to make herself into a force of nature that destroys witches before they're born.
Subverted? That girl doesn't fucking give up. She started as a Moe MoeShrinking Violet with a weak heart, weak eyes who had Madoka as her only friend. Then Madoka dies, killed by Walpurgisnacht. Little cute Homura wishes to go back in time to try to save her. As she lacks offensive magic of her own aside a weak laser-like thing, she starts building pipe bombs. It doesn't work as Madoka becomes a Witch after Walpurgisnacht is defeated. She tries again, and this time steals pistols and shotguns from the Yakuza, only to fail again as she has to kill Madoka herself after she used the last Grief Seed on Homura in order to let her go to the past again to try to save everybody. Instead of buckling under the pressure, Homura resorts to use a cold mask and military gear to try to take down all Witches, including Walpurgisnacht, alone. She fails, and witnesses Madoka one-shotting Walpurgisnacht and becoming a Witch capable of destrying Earth in ten days. When, that doesn't work, she tries again. Her psyche is barely holding together, she shunned everyone who used to be her friend, but she still doesn't give up and goes against the Nigh Invulnerable Walpurginacht again with hundreds of rocket launchers, mortars, a full gas tanker, anti-ship missiles and tons of explosives that she stole herself from the JSDF and the US Army. If this isn't a Determinator, then half of this page doesn't qualify either.
Pumpkin Scissors: Alice L. Malvin can't stand injustice or inequality - so much so that she will literally jump into any fight and won't back down, no matter what the odds are. This usually means her troops have to end up dragging her out of situations. There is an upside to this though - she's definitely not just the Idiot Hero when she gets serious.
RahXephon's protagonist, Ayato Kamina (no, not THAT one!) plays this trope straighter than a ruler. At one point, upon being told it'd take a miracle to save someone, he accepts it as a challenge and pulls it off!
Ranma ˝: Pretty much every character. Hell, Rumiko Takahashi is FAMOUS for making characters like this. It'd be quicker to list the ones that AREN'T determinators.
More in the manga than in the anime, Kenshin Himura defeats enemies seemingly by pure willpower alone, such as his determination not to kill. He survives several seemingly insurmountable fights in quick series, sustaining massive injuries each time and still managing to win every single time.
The young man who gave Kenshin the first slash of his X-shaped scar, Akira, did so despite being nowhere near Kenshin's level as a swordsman. Kenshin told witnesses, "His skill was nothing. But his desperation to live... was incredible." It didn't save him.
Sailor Moon: Usagi/Sailor Moon. Even when all her friends are dead and she's all alone against impossible odds, she keeps going.
Saint Seiya: Pegasus Seiya is one of the finest examples of this trope, if not the epitome of it. He gets dunked on his head in nearly every fight he puts himself into; he's taken punches thrown at light-speed with his face; he was once stomped so far into the Earth that it might as well have been a burial; and oh, he gets knocked off a cliff once every arc/movie. And he still manages to stand up tall at the end of each and every encounter. Somewhat subverted in the conclusion of the Hades arc, where he gets stabbed in the heart with a sword by the hands of Hades himself, and the blow was laced with a curse that's currently keeping him comatose. But chances are good that he'll be back.
In the prequel manga, Saint Seiya: The Lost Canvas, we have several examples among Athena's Saints and also among the Specters, but there are some people who deserve special mention:
Sagittarius Sisyphus. He had blinded himself, his body was broken by the fight against Aiacos and he had just ripped his own heart out to open the Gate to the Lost Canvas... and he was still able to join Shion and Regulus in launching Athena's Exclamation.
Radamanthys of Wyvern, who takes this to ridiculous levels. Relying on his Undying Loyalty to Hades (and Hades alone), he lives through: all the attacks of Scarlet Needle, being burned alive from the inside, ripping his own heart out, all the strength of Leo Regulus], Regulus' Athena Exclamation, Regulus' Zodiac Exclamation, which has the power of all TWELVE GOLD SAINTS, and Regulus punching a hole through his chest. What finally kills him isbeing split in half by Alone... But not before he destroys Athena's picture, restoring her power and turning the tide of the Holy War. If this wasn't enough to earn him his Determinator status, the fact that, after doing all that, he still was alive enough to save Pandora from Alone's attack and teleport her away from the Lost Canvas probably does it.
Saiyuki: The Sanzo-ikkou. Throw anything at these guys, anything, and they will get up, kick your ass, and continue West. Godlike powers, weapons that neutralize existence, whole armies of demons, or just filling them full of lead...whatever. And it's not because they're heroic or have ideals to live up to, it's just because they're adamant about setting their own path and too damn stubborn to quit. Ever.
Samurai 7: Kyuzo is willing to destroy as many enemies as necessary just so he'll be around for the one duel he actually wants to fight. After destroying countless mecha and getting his arm severely damaged, he responds that he can still fight with the other one.
'Kazuma: "I'm not quitting. Even if I fall, I'm gonna fall forward! Right Kimishima?"
Shaman King: Subverted. Yoh Asakura, after ten or so volumes of trying to become the Shaman King, gives up without a fight. Granted, it was to bring Ren back to life, but he didn't even think twice about it.
Skip Beat!: Kyoko Mogami has this as one of her defining characteristics, (the other being holding intense grudges against anyone who mistreats her). In a flashback in Act 7 of the manga, she is given a daruma doll as a gift from her old boss, which is a symbol for unwavering persistence.
Slam Dunk: Most (if not all) of the teams, but in particular the main protagonist Sakuragi Hanamichi, who helps will his team to victory against all odds and a nigh-invincible opponent, despite crippling pain from a potentially career-ending back injury.
After the 75th floor boss battle in Sword Art Online, when Kirito is fighting Akihiko Kayaba, he is actually killed with a sword through the chest. Then he stabs his opponent fatally, winning the game and saving his own life. A few minutes later, he wakes up in the real world, his body weak from two years on life support, and immediately goes looking for Asuna.
Tekkaman Blade: D-boy lives on this trope. Early in the series when his crystal was disabled, he still kept going until it actual broke. In the next episode he took on a Tekkaman with no powers whatsoever. In the final chapter, his Determinator status gave him the means to achieve Unstoppable Rage.
Noal: "He fights Radham with his human body. Geeze, he really is a dangerous guy!"
Kamina inspires the Badass Crew to fight even when they have no chance of winning. Being blasted apart by the pure awesome he radiates isn't worth missing the show for when he starts to fight. His line early in the series, "A real man never dies even if he's killed!", sums up this trope as perfectly as a single sentence can. Considering the posthumous influence his attitude and ideals enjoyed, both in the series and in real life, it seems he was right. Simon also becomes just as much a Determinator as time goes on.
And if you do think about the first one to three episodes, even Kamina's massive determination was inspired from his dead father.
More or less how it was spread: Kamina's father=>Kamina=>Simon=>everyone and anyone else.
Of course, due to the way Spiral Energy works, sheer willpower and fighting spirit can make the impossible happen. Hell, Lagann uses it as fuel.
All of Kamina's catchphrases are pure Determinator stuff: "Kick the reason to the curb and do the impossible!", "Who the hell do you think I am!?", "Don't believe in yourself, believe in me who believes in you!" The whole anime is about human determination to overcome the obstacles. Every episode, the characters confront a seemingly impossible obstacle, then Kamina/Simon refuse to surrender and somehow overcomes it performing impossible feats it by sheer determination, only to discover a new obstacle that dwarfs the previous one, then they perform and EVEN MORE IMPOSSIBLE FEAT, and so on, in a continous excalation. Even the two recurring themes of Spirals and Drills purposedly remind the concepts of excalation and destroy obstacles, all driven by determination.
Hell, when they talk about the heroes on the Deteminator main page they say "For heroes, "hit the villain until he drops" is a virtue that will be rewarded in a Heroic Second Wind moment, especially if they're the Badass Normal with a Screw Destiny attitude." This describes Kamina's last Battle to a T. Willing himself back to life with the Heroic Second Wind and owning the Big Bad General who has been unharmed so far with one attack. Not to mention that despite how awesome he is, his spiral power is average, qualifying for Badass Normal. As for the Screw Destiny part... Do I really have to say it?
He does admit in-series that a great deal of his hot-bloodedness is faith in his allies and exceptional bluffing skill more than actual confidence in himself. Word of God in one of the supplementary art books comparing characters' relative Spiral Energy levels actually pointed out his is quite low compared to most of the rest of the cast, due to his hidden lack of confidence in himself. It's his ability to inspire others that makes him a great leader and team player.
Simon's talk with Possessed Nia when she tells him there's a less than .01% chance of his success. "Maybe, but it isn't zero. And for me, that makes it the same as a 100% chance!!!" Distilled trope, thy name is Simon.
And later, they succeed at something even when the chance of success was zero. "It seems that theoretical calculations are pointless with you."
Tiger & Bunny's Kotetsu/Wild Tiger will fulfill his duties to protect the people in all circumstances and at all cost. This is the case even when his powers start to decline. Rather than find new work, he decides he may as well become Sternbild's first Non-Powered Costumed Hero. Barnaby also proves to have a mile-streak of stubbornness, albeit a far more self destructive one. Most of his life was spent on a single-minded, twenty-year quest to find his parent's killer at the expense of anything resembling a well-adjusted life.
A Certain Magical Index: Kamijou Touma. Burn him with fire of about 4000şC, cut his arm off, crush him with I-Beams, send a giant golem, call an angel or fight him using the power of God's Right Hand, Archangel Michael and he won't step back, he will keep going on to acomplish his goals.
Is it any wonder he's called TouMAN by the fandom?
On at least one occasion Stiyl has also demonstrated this. Despite being having exploded into what was described as a constellation of organs and viscera that filled an entire room, he not only remained conscious and aware but also managed to perform magical spells and reassembled his own body.
Starscream from Transformers Cybertron has many determinator moments, but his greatest is when he takes on the entire Autobot team, getting directly hit by most of their most powerful attacks before defeating them. Visibly grievously damaged (in a show where most characters rarely show visible damage) he then takes on Optimus Prime who is at full power. He is briefly knocked unconscious, but revives in the nick of time to launch one final cheap-shot and ultimately achieve his objective and acquire the Omega Lock.
Starscream: "True power is simple determination, making the impossible possible through sheer force of will. I have risked everything in pursuit of this power! I have given myself over to it, body and spark! Now witness...the power of one who has sacrificed everything!" (Battle Aura flares up)
Also, Kurogane. He'll do anything to keep Fai from dying, even if it costs him an arm. He's also quite willing to go against Fai's wishes on this.
Oh so much determination in the second season of Uchuu Senkan Yamato - the Argo has been badly damaged, the Wave Motion gun inoperable, the Earth Defense Force has surrendered to Prince Zordar and the Star Force only has a couple of fighters and a handful of missiles left to their name - and yet they fight back, drive the Comet Empire Citadel off earth and manage to save the day.
Ataru Moroboshi is a rather luckless, perverted version of this trope. Heaven help anyone between him and a beautiful girl, or the girl herself if she tries to use him to get at Lum and then underestimate how far he'll go for a kiss or grope. Nothing will stop him—common sense, decency, the law, fate, the gods, beatings, near-electrocutions, explosions, military-grade weaponry, armies, you name it. Even evil spirits are scared of him. He demonstrates a tenacity that makes his "friends" wonder if he's even human.
It even plays into the end of the series—Due to an odd series of events, Lum loses her horns temporarily, and they have to repeat the pilot episode's chase scene. Except this time Lum is tired of Ataru's refusal to commit to her, and is bouncing between flying well out of reach and hiding in orbit, refusing to come down until he admits he loves her—if he doesn't by the end of the time limit, the Oni will use their technology to wipe the memories of them from Earth's populace. At the end, after days of constant work, not sleeping nor once giving up, Lum is openly enraged, asking him "Do you really want to forget me?! Just say it!" while Ataru is constantly chanting "I won't." In the finale, it's revealed that he's really chanting "I won't forget you," and he was holding onto the horns she had previously lost as a keepsake—when he screams the previous line out, he falls to his knees and accidentally drops them, breaking down in the process. This is touching enough that she picks him up as he collapses from exhaustion and places his hands on her horns, finishing the game for him.
In a way, Mr. Fujinami is a huge parody of a determinator. He wants a son and therefore refuses to demonstrate any acceptance of the fact that his Daughter isn't a boy in the hopes that reality will eventually be bent by his determination. Well except for the two times he's offered genderbending technology.
Vinland Saga: Thorfinn will do whatever is required to earn a duel with Askeladd, even if it's down right suicidal.
Windaria: Marie, who defies death itself to wait for Alan to return.
Witchblade: Masane Amaha was mostly a Punch Clock Hero, at first. However, since circumstance (being a single mother) forced her to do something to provide for her daughter Rihoko, she willingly put herself in harm's way for said daughter. In fact, she refused to DIE one episode because no one would be around to effectively parent her child. Once that wasn't a problem anymore, then she WAS willing to die.
Like a total Badass with a smile on her face. Basically, Rihoko was the one motivation that kept Masane from refusing to give up. Period.
Xabungle: Jiron Amos is possibly the most hot-blooded Real Robot pilot ever. The guy has the guts to take on, catch, and throw back an ICBM, isn't scared to take on the entire military force of the Innocent, and ignores the statute of limitations on his parents' murder so that their murderer is killed.
Watanuki of Xxx HO Li C proves to be incredibly stubborn, in a rather dangerous sense. The earliest example of this that stands out is when he starts visiting a woman he befriended, every day, without fail, because he doesn't want her to feel lonely. Even when he learns that being in her presence is slowly killing him. Later it even becomes common for him to assert in a bad situation that he won't die, because he'll "survive through determination".
Katsuya Jounouchi, proven when he got all the way to the semi-final of Battle City without holding an Egyptian God card. He was almost about to defeat the Big Bad in that duel, but he collapsed after withstanding an attack that could've killed him. He lost by default. This was only one of two times said Big Bad actually showed FEAR, the other being his eventual defeat. That's right, Jounouchi, the normal guy with no magic powers or God Cards SCARED Yami freaking Malik.
YuYu Hakusho: Kuwabara tends to win all of his major battles by refusing to stay down. One example is his battle against Rishou in the Dark Tournament. Kuwabara starts the fight severely weakened from a previous fight and proceeds to get the stuffing beaten out of him for an entire episode. In the end, he manages to pull off a win with some help from The Power of Love. Yusuke and Kurama have had their Determinator moments as well.
From the same arc, Byakko. Kuwabara blowing him up? Just a flesh wound. Kuwabara throwing him into boiling lava? That hurt and actually crippled him, but he would have recovered if Seiryu had let him borrow a a little yoki. But Seiryu blasted him into pieces... And his now frozen head still asked why he did that.