Son Goku from Dragon Ball. He always takes down the big bad of the arc(with the one exception of Gohan defeating Cell in which case he was a supporting role), and never ceases to continually get stronger. Many of his enemies become his friends after he defeats them. He's also very naive, qualifying him for Idiot Hero. He says that he does not want to be continually defending Earth, hence why he first trains Gohan to be the one to protect earth, and when that fails, he later trains Uub.
Shaman King: Yoh Asakura is the main good guy shaman. At one point, Anna compares him to a traditional knight hero going to defeat the evil king and save the princess. Except the princess is his evil Big Bad brother Hao and the evil king is called 'Hao's hatred.
Yoh actually takes this trope so far that he's practically a messianic character. He's so good-hearted, selfless and kind that his example is able to change even the most evil people over to the good side. As a matter of fact, he does actually change almost every single villain in the series over to the side of good, except Zeke/Hao, because Zeke/Hao is just that evil.
Maka Albarn from Soul Eater fits this role well, and largely above all the other characters in the anime/manga. She is the only character of the main cast to have very noble and heroic personality traits, in addition to having the strong desire to help others and defeat evil.(See also The Leader.)
Yusei Fudo from Yu-Gi-Oh! 5Ds and Yugi Moto from Yu-Gi-Oh! fit this role well. The main protagonists of the other two series', Yuma Tsukumo and Jaden Yuki, didn't quite have the heroic or noble traits to fit the role, though. They were more closer related to the Idiot Hero.
Monkey D. Luffy of One Piece fits this perfectly. He's an Idiot Hero, The Captain and The Leader of the Strawhat Pirates, and is always the one to take on the Big Bad in the end of every big story arc. His normal outfit, before and after the Timeskip, is even Red and Blue. In his own opinion, note chapter 634 in the manga Luffy has explicitly stated that he isn't, and doesn't want to be, a hero. In fact, he only agrees to save Fishman Island by being bribed with food. Then again, Luffy has never been a 'traditional' hero....
Tengen Toppa Gurren Lagann Kamina styles himself as the bold and charsimatc hero that will save the world from Beastmen. In a private moment with Youko, he says his dream is to create a world where children can look up at the stars without fear. Then he dies and Simon takes up the hero role after a Heroic BSOD. He slays the Physical God with a giant energy drill to save his princess.
Chrono Crusade: Rosette Christopher is front-and-center in The Team and the one that holds most of the group together.
Monster: Tenma spends the show hunting down a monster of a child to prevent him from hurting others.
Natsu from Fairy Tail. An interesting variation - Erza actually leads the team, and Lucy can be argued to be The Protagonist but Natsu is the traditional heroic figure.
The eponymous character of Black★Rock Shooter OVA is not an example of this trope. In fact, it's strongly implied that were she to follow this trope to the letters (always win, resolving everything with force) she will notget thegirl.
Kazuma Shudo of Kagerou-Nostalgia steps into this role as the story progresses, shifting from a bitter, disillusioned, and nearly psychotic Anti-Hero, to The Leader of the reincarinated heroes, and a more or less altruistic (if still incredibly cynical) character.
In High School D×D, Issei Hyoudou is the hero of the group and the one guy that holds everybody together despite the fact that he's an Idiot Hero who is only capable of thinking perverted things.
Death Note presents an interesting case of Playing with a Trope. While Light Yagami is clearly The Protagonist and believes himself to be the hero that will cleanse the world of crime, the opinons of others in the show are varied. Some agree and other think he's a serial killer no better than the people he kills. L, the detective hunting him down, is then the Hero Antagonist to Light's Villain Protagonist.
Vividred Operation: Akane Isshiki is the series' main protagonist, being a cheerful, hardworking and athletic girl.