You can’t have a serious discussion about characters who never, ever give up without talking about Roland Deschain.
The hero of Stephen King’s Dark Tower series, Deschain is the last of his kind, on an impossible quest to reach the Dark Tower and save not only his world, but all the parallel worlds held together. His journey began at the tender age of 14 when he was forced to fight his teacher, and only got more difficult from there.
The Dark Tower recounts aspects of Deschain’s past, and the immediate portion of his journey to the tower is a parallel to those darker deeds, and his chance to redeem himself. Of course, there are giant mechanical bears, sentient trains, poisonous crab-slug monsters, and various other bad guys (not the least of which is King’s infamous Randall Flagg) trying to stop him, and sometimes managing to take pieces of him–either in the form of fingers and toes, or in the form of a member of his ka-tet.
But he keeps going. He faces everything thrown in his way, venturing into other worlds and other people, losing so much along the way … and then he stands back up and keeps going. And when you consider the cycle of reincarnation that concludes the series, Roland’s journey has been infinitely longer and more treacherous than anyone else could have endured.
Time after time, he moves ahead. His fate is to quest for the Dark Tower, and even in death, Roland Deschain never, ever gives up the quest.