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.
Aoba Watase from Buddy Complex. Certainly more heroic than his buddy Dio. His story involves him saving a princess (Hina) from an evil king (Bizon).
Chrono Crusade: Rosette Christopher is front-and-center in The Team and the one that holds most of the group together.
Shido Itsuka from Date A Live. Despite his desire for a normal life, Shido will risk life and limb to save the Spirits and allow them to experience happiness.
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.
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.
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.
Tatsuya Shiba from The Irregular at Magic High School. He's the leader of his group of friends, more powerful than his peers, protects his sister with intense devotion and defeats the bad guys with little difficulty.
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.
The Macross series feature Love Triangles, usually involving two girls and one guy. That particular guy is usually the primary hero of the story. Unusually for a mecha anime, the hero is not necesarily the resident Ace Pilot.
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...
Saitama from One-Punch Man. He became a hero after his childhood dream to be able to beat bad guys with one punch. It turns out limitless power is "kinda boring actually". The intro theme for the anime is even titled 'The Hero!'
Re:CREATORS presents a variation. The story happens from Sota's point of view, but Celejia is the show's primary heroic character. As the show features fictional characters coming to life in real life, Celejia's story of origin is a straightforward heroic fantasy.
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.)
Lux Arcadia from Undefeated Bahamut Chronicle is defined by his righteousness and benevolence, to the point he destroyed the very kingdom he was supposed to inherit, simply because his father was an evil bastard.
D in Vampire Hunter D may appear like a dark and cold figure that slays monsters for money, but for most of the time he saves people and does the right thing without expecting a reward in return.
Variable Geo centers on Yuka Takeuchi, a part-time waitress/full-time martial artist. She goes on a search once she learns that her childhood friend, Satomi Yajima, has gone missing and eventually learns she's being manipulated by The Jahana Group.
Vividred Operation: Akane Isshiki is the series' main protagonist, being a cheerful, hardworking and athletic girl.
Yusei Fudo from Yu-Gi-Oh! 5D's 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. Though Jaden showed a bit stronger of this role during the 4th season.