Rainbow Dash: Today I learned what the most important quality really is; a certain kind of spirit; a stick-to-it-iveness. A never-give-up, can-do attitude that's the mark of a real winner! And this tortoise has it!
Prince Zuko from Avatar: The Last Airbender. Even before he knows that the Avatar is alive, he sails around the entire world looking for him. He then chases him from the South Pole to the North, surviving at least one assassination attempt as well as several duels, infiltrates two heavily-protected fortresses on the way, defeats a master firebender and battles with a master waterbender, and is only defeated when a blizzard prevents him from moving too quickly. He gets into about two dozen fights during that time... and that's just the first season. Sokka's opinion is that "if we know one thing, it's that Zuko never, ever gives up," but Zuko's statement is rather more poetic.
Zuko: "My father says [Azula] was Born Lucky. He says I was lucky to be born. I don't need luck though; I don't want it. I've always had to struggle and fight and that's made me strong. It's made me who I am."
His flashback in "Zuko Alone" shows just what the above line is all about. He was no prodigy like his little sister Azula, so his daddy Ozaipaid him virtually no mind and his little sister Azula laughed at his every misfortune even then. However, he always strove to improve his abilities and to prove himself to everyone. His mother recognized this, as shown by the following line, which came after the example in the flashback:
Ursa: "That's who you are Zuko. Someone who keeps fighting even though it's hard."
To a slightly lesser extent, Katara. Want to learn waterbending? Just a quick trip to the other side of the world. Your new friend's in prison? Okay, get yourself arrested and break him out. The local waterbending master won't teach girls? It's cool, just practice in secret, challenge him to a fight, and proceed to master the element faster than the Avatar. Want to help that repressed village? No problem, just disguise yourself as a minor deity and destroy a polluting factory. Oh, and about those guys who killed your mom... Not to mention several times she is all that's holding the team together, most notably in "The Desert". In her own words:
Katara: "I will never, EVER turn my back on people who need me!"
Fire Lord Ozai also has shades of this. It does not matter how freakishly powerful the Avatar is, the man simply will not admit defeat. Even when Aang actually took away his firebending and Ozai could barely move, he still conjured the effort to insist on his title as Phoenix King. Getting thrown in jail at least managed to humble him a bit.
The Legend of Korra: Tenzin (must run in the family, considering Katara's his mother); it took the combined might of the Red Lotus (four of the most powerful Benders in the world) to defeat him in Season 3, and even then he refused to stay down.
Tenzin: "As long as I'm breathing, it's never over."
Clarence: In "Sumo's Boat", Sumo spends the entirety of the episode building a boat, ignoring his friends, working through rain and storms, and working throughout night and day.
Optimus Prime of Transformersis this trope, with Megatron coming in a close second.
This is a pretty regular thing for Starscream. In Transformers Generation 1 he tried to take over the Decepticons at every possible occasion—even after he died and came back as a ghost.
Dinobot from Beast Wars almost never willingly runs from a fight. This reaches its climax in 'Code of Hero', when his insistence on stopping the Predacons from destroying the early humans costs him his life. Depth Charge also fits this trope, as his persistence to find and destroy Rampage leads to him putting that mission above all else, including stopping Megatron from destroying the Autobots and changing the universe as we know it. Really, Depth Charge needs to get his priorities straight. By the end of it, he was on board to stop Megatron. Unfortunately, at that time, Rampage was sent out to stop him.
In Cybertron, Starscream was so much of a Determinator that he managed to use Heroic Resolve in a way that's usually reserved for... well, the heroic. At one point, he took on the combined might of the entire Autobot team to get the MacGuffin, dragging himself onward no matter how much damage he took, and taking the victory in the end with one well-timed dirty trick (hey, he is Starscream.)
In Transformers Animated Starscream kept trying to kill Megatron after dying normally, becoming effectively immortal, dying many times, becoming a head, and then being killed again.
Waspinator becomes this in Animated. Determined to get revenge on Bumblebee, he'll get blown into pieces to get the justice he thinks he deserves.
Kyle is this frequently, almost always in an attempt to thwart Cartman's latest scheme.
Cartman may be a gigantic prick, but when he wants something he'll go to almost ridiculous lengths to get it.
Butters in "Super Fun Time".
Butters: "Teacher, my partner is back on the bus."
Jimmy in "Erection Day" and "Up the Down Steroid".
Dib of Invader Zim is a Determinator well past the point of insanity, considering Failure Is the Only Option and he's surrounded by people ignorant of problems. Zim is a similar case, continuing his "mission" even after being exiled due to his entire species loathing his very existence, although there's disagreement as to whether he merely ignores their scorn or he really is that stupid.
Riley from The Boondocks. He's not a very good fighter, but he's able to take a lot of punishment before going down.
In the Season Finale of Justice League Unlimited, with no time to come up with a better plan, Batman keeps on coming after Brainiac-Darkseid with nothing more than his fists, feet, and utility belt, relentless even though the other tosses him away casually. After the last such attempt, irritation is evident.
Brainiac-Darkseid: You still try to fight? tosses him thirty feet Can't you see that it's hopeless?
Batman also provided a page quote for the main Determinator article:
Dr. Destiny: But you're different. You don't have any special powers.
Batman: Oh, I have one, Johnny. I never give up.
Just ... Batman. Period.
Earlier in the series was another example. Shining Knight, an embodiment of the Knight in Shining Armor trope, was fighting a mutated and transformed General Eiling and refused to give in no matter how badly he was beaten.
Then there's Wonder Woman in the adaption of For the Man Who Has Everything. She absolutely gets the shit beat out of her by Mongul for almost the entire episode and refuses to stay down even though each hit she takes brings her closer and closer to death. She shows that you CAN be on the receiving end of a total Curb-Stomp Battle and still look Badass.
And, after this, she gathered all her remaining forces to free Batman from Black Mercy, resist to the plant's attempts to dominate her, and throw it at Mongul.
Captain Atom in Flashpoint. He gets a horrible ass whooping by Superman when trying to keep Huntress from helping the previously tortured Question escape Cadmus. Though Supes is perfectly fine at the end of the Battle, Cap is beaten so badly his suit cracks and he's barely able to walk, much less throw a punch. And yet he still keeps coming until he's completely unconscious.
Superman:You fought a good fight, stay down!
Captain Atom: ...I can't do that, Superman. (rushes him)
Superman:(dodges his attacks and punches Captain Atom into the wall)
Considering the fact he stayed locked up in prison for twenty years and, upon escaping, was just as intent on becoming the Dragon Warrior as ever, Tai Lung from Kung Fu Panda probably qualifies. As does Po himself. Even when Shifu does his best to make Po quit, Po never does. While Po is in the middle of rolling down the mountain due to a kick from Shifu, the other students have the following conversation about it.
Tigress: If he's smart, he won't come back up those stairs.
Monkey: But he will.
Viper: He's not going to quit, is he?
Inverted when Po does quit, Shifu is shocked when he didn't quit before, why he was quitting now when Shifu needed him to be the Dragon Warrior.
Po: From the first moment I got here, you've been trying to get rid of me!
Shifu: Yes! I was! But now I ask you to trust in your master as I have come to trust in mine.
Po: You're not my master. And I'm not the Dragon Warrior.
Shifu: Then why didn't you quit? You knew I was trying to get rid of you, yet you stayed!
Po: Yeah, I stayed. I stayed because everytime you threw a brick at my head or said I smelled; it hurt, but it could never hurt more than everyday of my life just being me. I stayed because I thought if anyone can change me, can make me not me, it was you! The greatest kung fu teacher in all of China!
Darla Dimple from Cats Don't Dance, while not physically intimidating, is still a Little Miss Badass who refuses to allow anyone to interfere with her films, to the point of constructing an Evil Plan to discredit them, and (almost) suicidal attacks on their shows.
Darla: I didn't get where I am today By letting myself get pushed around! And no man nor beast or kittycat or doggy Is goin' to drag me down!"''
Also, Danny. Despite all of the hardships and prejudice he goes through, he still doesn't give up on his dream of being famous.
All of the animals that stayed, as shown in the final big musical number "Nothing's Gonna Stop Us". Surprisingly, Darla keeps fast to her perseverance in trying to stop them during it, clawing her way up the backstage wall to reach The Grandaddy of All Switches despite having been thrown about, electrocuted, and falling from high places.
Phoebe: C'mon, Arnold, let him go. It's just an egg.
Arnold: But it's Dr. Skeledon's egg. And it's up to me to get it back to her!
This episode also ended with Arnold fighting off a T.Rex to protect the class and get the egg back.
Parodied, like many other things, in an episode of Family Guy.
Chris: Dad...I want to quit the Boy Scouts.
Peter: Chris, you're a Griffin! And the Griffins never quit!
Cut to scene in hospital. Peter is in surgical scrubs and holding defibrillator paddles.
Peter: Clear. poomp Clear. poomp
Patient: Thanks, doc, you saved my life!
Peter: Clear. poomp
Another example comes from Carol Channing, of all people. During her celebrity boxing match with Mike Tyson, she just WOULD NOT go down.
Courage the Cowardly Dog will brave any danger and take any punishment to save Muriel. His best example of sheer determination didn't even involve saving her. After getting beaten to a ridiculous pulp by Mecha-courage repeatedly, he just keeps getting up, eventually shorting out the robots batteries through sheer persistence.
Courage routinely lampshades this trope with one of his catchphrases
This is also shown in the episode "Ball Of Revenge," when he refuses to give up against the villains to save Muriel.
The Simpsons: Apu Nahasapeemapetillion thinks that the town of Springfield should hire more police officers. In one episode, he mentions that he'd been shot by armed robbers 8 times already this year. As a result, he almost missed work. Almost.
When Bart was playing hooky, he destroyed a wooden bridge to stop Principal Skinner from following him. Skinner simply walks into the river, momentarily emerging at the other side.
In fact, both Homer, Bart, and especially Lisa exhibit this trait when they're passionate about something they care about or have their hearts set on something they believe in, while Marge exhibits these traits for the sake of her own motherly propaganda, usually acting as a foil for her husband and children, by interfering with their interests or trying to outlaw something that they happen to love.
Wile E. Coyote, Sylvester, Elmer Fudd, Yosemite Sam, and Marvin The Martian of Looney Tunes.
No mention of Daffy? He'll do whatever it takes to upstage his rival.
Finn's unwavering moral code is what makes him such an awesome character. No matter how often he is criticized for it, or how many times its challenged, he is freaking intent on doing the right thing even if its the wrong thing by accident.
Simon Petrikov is so absurdly determined as to make Finn look spineless. There is an Artifact of Doom which, when used by an alternate timeline Finn, caused full and complete corruption in seconds. Simon, in contrast, had the artifact for decades before he lost his sense of self - and even centuries later, he's still not fully corrupted.
Monkey in the Dexter's Laboratory 'Dial M For Monkey' short series. Most notable in one episode where Monkey fights against intergalatic wrestling champion Rasslor, and gets his ass handed to him over, and over, and over, and over, and over, and over...until Rasslor finally throws the match out of respect for Monkey's inability to stay the hell down.
Rasslor: This is unbelievable! I could crush your body, I could smash your bones, but I could never break your spirit!
Phineas and Ferb never give up on their attempts to do the impossible. Also, Candace, Isabella and Doofenshmirz qualify
Total Drama Island gives us Ezekiel, who, after being kicked off first once more in the third season, managed to get back in the plane four times after Chris tried to get rid of him, eventually becoming some sort of zombie. All to prove he was worthy of winning the season.
In the Spider-Man (1967) episode, "The Sinister Prime Minister," Spidey is the only one who knows that a foreign Prime Minister has been abducted by a impostor and must strike again and again in order to rescue him.
Tunar: "I'll follow ya straight to the flamin' pits of Magmel before I give you up!"
Series' Hero Lion-O also has this quality, though to a more benificent end. In "The Duelist and the Drifter," after suffering defeat in a duel with a predatory Master Swordsman, Lion-O heads to a forge and stubbornly insists that he'll make a sword that can withstand a rematch, no matter how many attempts it takes, despite having no training in swordsmithery. A retired swordmaker is so amused by Lion-O's determination that he hangs a lampshade on it, and offers to lend a hand.
The Swordmaker: You don't give up easily, do you?
Even the villains notice it:
Kaynar: You're like a toy that never breaks, I LOVE IT!
Hell, even death and failing the trials to come back to life wasn't enough to stop him.
The six main ponies from My Little Pony: Friendship Is Magic are each considerably persistent characters, and the value of perseverance is a common theme throughout the series.
Applejack is a surprising deconstruction, though, in that in the fourth episode of the first season she is used to demonstrate that persistence can be a bad thing if one is overly persistent in trying to do something when surrounded by evidence that it cannot be done; in that case, trying to pick all the apples in the orchard herself and becoming sleep-deprived in the process. Of course, this is presented in the context of being compared to doing something else that would achieve the same thing more practically; namely, accepting help from friends in picking the apples.
The tortoise from "May The Best Pet Win" takes this arguably even further; taking part in every competition even after losing each previous one, and never giving up on winning over Rainbow Dash's approval even as she keeps putting him down.
Princess Cadance has also proved herself to be capable of this. During her opening episode, "A Canterlot Wedding Part 2", she runs past a momentarily hesitant Twilight to push a mine cart free, despite having been without food and water for days. She momentarily collapses, but partially because she physically has run out of energy. In "The Crystal Empire", she maintains a force field to keep Sombra out for three days, nonstop, again without much food (or sleep, for that matter).
Saint Walker from Green Lantern: The Animated Series. Climbs a mountain in search of a savior to battle the Red Lanterns, and when he reaches the top he seemingly finds nothing there. Instead of giving up, he keeps hope and vows to find a way to battle the Red Lanterns, resulting in the Blue Lantern of Hope to come to him, its reflection showing him to be the savior he was searching for, and give him the first Blue Lantern ring.
Mr. Bogus would also qualify for this trope. Despite the fact that he is only a few inches tall, nothing will stop him from fixing a problem that has arose in either Tommy's house or in Bogusland.
Everyone from Animaniacs shows this, but one interesting example is Yakko Warner singing all the words in the English language. He's obviously broken down when shown singing the "Z" words but refuses to quit and doesn't faint until being requested to sing all the numbers above zero.
Eddy from Ed, Edd n Eddy. When he has his mind set to a goal, he refuses to give up (due to his stubbornness, but still). A perfect example for this would be in the episode "The Good, The Bad, and The Ed."
The Kankers, when it comes to the Eds.
Red Hood, of Batman: The Brave and the Bold suffered the same chemical bath as the Joker. However, he didn't crack, and he kept this up throughout his career, repeatedly battling whole teams of villains with nothing but throwing knives and grit. His response to electroshock torture? "Ahaha, it tickles!"
Wicket from Ewoks. Not so much in the first season of the animated series and the comics, taken to the extreme in the second season; where he's often motivated solely for wanting to fill his belt of honour.
Many characters in The Avengers: Earth's Mightiest Heroes, but especially Captain America and Ms Marvel. As far as they're both concerned, this is humanity's hat, and the Kree and the Skrulls definitely understood this after they were through with them.
(To Vision) Captain America: You think I'm fighting for me, to save my own life? That's why you'll never win. I'm not fighting for me. I'm fighting for them! And that's something you'll never understand, machine! It's the very thing that makes us human.
(To Skrulls) Captain America: There's more to human beings than our bodies and minds— something you'll never understand. Our spirit! We never surrender! We never give up!
Zig from Zig And Sharko never gives up on trying to catch Marina The Mermaid. Even with Sharko protecting her.