Listen. I'm about to suggest something that you'll probably struggle against. You might chuckle or roll your eyes. If so, stop it.
No, wait, really. Is it silly? Yes. Are the super powers crazy? Oh, without doubt. However, it's also brilliant. It's the most epic work of fiction I have ever encountered, and I do not say that lightly. Like Dune or A Game of Thrones, the scale of the author's vision is massive and astonishing. It's not like Bleach, where you get the feeling the original story ran its course long before the end and all the rest is just tacked on. With One Piece, you'll get references made FIVE HUNDRED EPISODES before. I've read hundreds of books, reading fantasy and science-fiction was always my primary hobby, but the scope of One Piece leaves me in awe. Even beyond that, the ability of the author to stir emotion is remarkable. For cat's sake, there was an episode where I cried like a little girl over a ship. I mean tears running down my chin crying--and no, I'm not ashamed of that, it was just that damned powerful.
One Piece is packed to the brim with epic.
Anime with fighting or conflict:
Baccano! - Excellent story with a LOT of characters. Involves immortals and psychopaths and lots of blood.
Fate/Stay Night - Saber is one of the most awesome characters ever.
Gurren Lagann - Another anime that screams epic. Doesn't really kick in until around episode 8, but when it does, it gets real in a hurry. I almost dropped it until a certain moment; after that, I was completely hooked.
Kara no Kyoukai: Garden of Sinners - Absolutely amazing. Character depth, plot twists, incredible production values. I mean, seriously, the main character might be a serial killer, you don't even know.
Monster - A legendary psychological thriller that will have your head spinning. It's about a doctor who saves the life of a psychopath, a boy who grows up to be a monster, but it's so much deeper than that.
Samurai Champloo - Offbeat action anime with two swordsmen who are exact opposites on a journey with a girl. It's good.
Steins;Gate - This is all about time-travel, paradox, and wide-spread conspiracy. A fantasic anime.
Welcome to the NHK - Everyone should watch this, anime fans or not. It's flat-out brilliant, as good a series as I have ever seen, animated or live-action. It's completely hilarious while being very powerful and profound. It deals with personality and social disorders/problems like addiction, paranoia, depression, and suicide. The closest thing I've ever come across like it is Catch-22.
Skip Beat! - It's about a girl who gets dumped by the guy she loves and realizes that he'd been using her all along until he found success as a singer/actor. Yes, it's got a bit of romance, but it's mostly about a person who pulls herself up by her boot straps and refuses, absolutely refuses, to give up even when life keeps knocking her down.
Nodame Cantabile - A really, really offbeat romantic comedy. The girl is a complete ditz/slob and the guy is rich, arrogant, and a genious at music. Somehow it works. Nodame, the girl, is just so unique, such an original character, that she grabs your attention.
I'd better stop here before I get carried away.