I'd like to read a manga where the protagonist is a loser at the beginning but slowly gets better through hard work and training. No sudden awakening of magic powers, sealed demons (lookin' at you, Naruto) or other shortcuts - they're just normal people working hard to achieve a goal. It would be great if they still failed from time to time. It doesn't have to be a fighting manga, anything is fine with me.

One-Punch Man

Hajime no Ippo and Shijou Saikyou no Deshi Kenichi?

~4400 manga completed
~1700 ongoing...

Slam Dunk is a really great manga like this. It's about basketball, but you definitely don't need to be into basketball to enjoy it (I am not). Shounen.

Gantz, more or less. There are a couple times when they get better weapons, but it's mostly about the main character becoming less of a chickenwuss and learning how to fight. Seinen.

Ashita no Ousama if you're up for josei - this is a little different, it's about a girl learning to be a stage director, so it's more about learning skills than training per se.

Bakuman - Again nothing physical about this one, it's about aspiring mangakas. Shounen. Compared to the other three, there's not as much failure, but there is a bit.

most shounen sport manga is like this. some of them are:

Hikaru no Go - there is a spirit hanging around him, but all his victories come from his desire to really learn the game
Rookies - a hot blooded delinquent sport manga
Baby Steps - the very few shounen sport protagonist that use calculation and info gathering as his tools to win
Ping Pong - very intense, but the characters are more important than the sport
Eyeshield 21 - it can be quite fantastical, but the mc really do work hard

great fighting manga
Holyland - one of the few realistic fighting manga, with tips to the readers.
All Rounder Meguru - another great realistic manga, though the emotions are a bit muted.

great manga about growth
Beck - the growth of singer/songwriter/guitarist
Ashita no Ou-sama - the growth of a playwright

manga about fighting in fantastical setting
Mx0 - unique that the mc is great because he dont have power others have and have to use his brain
One Piece - adventure that brings gradual growth. i dont think the devil fruit is special since most people seems to have it
Hunter x Hunter - mc grow with each adversity
All You Need Is Kill - mc have to die to gain experience

that's all i can find. try them, they are great!