Dragon Ball: Do I need to explain it to you? Its freaking DBZ!
Rurouni Kenshin: A personal favorite. It is 1878, a peaceful time after the troubles of the Meiji Restoration and the Satsuma Rebellion. Kamiya Kaoru, the spirited and young head of the Kamiya Kasshin kendo school, is having trouble fending off a couple of no-gooders that want to take over the dojo. Through sheer luck, she gets a vagrant swordsman she mistook for a killer to help her against the baddies.
To her great surprise, it turns out that her initial assessment of this vagrant wasn't, strictly speaking, mistaken after all - the swordsman turns out to be Himura Kenshin, who used to be known as the Hitokiri Battousai ("the quick-drawing assassin") during the turbulent years of the Meiji Restoration when he acted as an assassin for the revolutionary anti-shogunate forces. Now he atones for his past sins wandering around and righting wrongs without killing anyone.
Later on, the Kamiya Kasshin dojo adopts two more hangers-on: Sagara Sanosuke, a brash youth with spiky hair and phenomenal physical strength; and Myoujin Yahiko, ex Street Urchin who evolves into a Kid Samurai. We also have Takani Megumi, a beautiful woman who knows her medicine quite well and mothers them all. This is good, as Kenshin's foes from his murky past tend to turn up with distressing frequency.
Buso Reiken: Busou Renkin (literally "Arms Alchemy") follows Hero and Ordinary High School Student Kazuki Muto, who one night is killed trying to protect a mysterious girl who was being attacked by a monster. However, to his surprise, he wakes up again the next day, seemingly good as new. It turns out it wasn't a dream either when the girl, Tokiko Tsumura, shows up again, informing him that she felt responsible for his death and resurrected him with the power of Alchemy. From here on Kazuki discovers his own powers and, together with Tokiko, journeys into the world of Alchemists and their Renkin as they fight the monsters known as Homunculi. No, not the Fullmetal Alchemist ones, though there is definite overlap between the series theme wise.
A manga by Nobuhiro Watsuki, the creator of Rurouni Kenshin, it was later adapted into an anime and a light novel. The series is in many ways a parody of the shonen genre and hence often has tongue planted firmly in cheek. And when it's not being an affectionate parody, it's deconstructing various Shōnen tropes, particularly the Shonen Upgrade.
My personal Favorite: + Anima: A series written by Natsumi Mukai, and licensed in America by TOKYOPOP. Beings who possess animal-like powers walk among humans in this alternate universe. These mysterious mutants, the +Anima, are shunned by society. Four outcasts in particular - Cooro, a boy with crow-like powers; Husky, a fish-boy; Senri, a bear +Anima; and a girl named Nana, who wields the power of the bat - search for others like themselves while trying to gain acceptance in a world cruel to anyone or anything that is different. Think of it as like X-Men, only more adorable.
AKIRA: In July of 1988 (or at 2:17 P.M. on December 6th, 1992), a mysterious black-domed explosion destroys Tokyo and sets off World War III. Thirty-one (or thirty-eight, depending on whether it's the manga or the film you're dealing with) years later, the rebuilt city, now known as "Neo Tokyo" has fallen into decay.
Two rival biker gangs, the Capsules and the Clowns, are having a turf war one night, when one of the youngest Capsule members, Tetsuo, almost literally runs into an escaped government test subject. Moments later, the test subject is taken back into custody by the army. However, they also decide to take Tetsuo with them. He then becomes the newest test subject for the "Akira Project." But when Tetsuo's powers awaken, the combination of an inferiority complex harbored since childhood with power beyond Tetsuo's wildest dreams waste no time in driving him insane. He escapes the lab and goes on a super-powered rampage through Neo Tokyo, killing and destroying everything in his path. It falls to a handful of people, including Capsule leader and Tetsuo's friend Kaneda, to put a stop to the destruction.