Good "real" fight scenes
 

I'm looking for some anime with good fight scenes. No super powers or magic though. This mostly limits it to martial arts and sword fights. One key thing is very well animated fight scenes.

From what I've seen already:
Sword of the Stranger
Hajime no Ippo
Samurai X
Baccano! (some supernatural stuff but not too much)
Cowboy Bebop
Samurai Champloo

History Strongest Disciple Kenichi- Kenichi vs. Hermit (2nd meeting).
Full Metal Panic: The Second Raid- Mao vs. Xia Yu Lan.

Bamboo Blade - fights with bokkens (bamboo swords), about 3 Tomoe fights with great animation
Black Lagoon - fights with guns
Wolf's Rain - excellent animation, fantastic action, but they're not "fight" scenes since a lot of times its wolf vs robot, wolf vs humans

I second this, the main character has great development, starting as weak and useless but gradually becoming stronger and stronger through hard work. Most of the fights I found short and technical, in a good way. There are a few people who exhibit what may seem to be a "power" but generally turns out to be a technique of some kind.

The end it does start go a little more over, but not nearly as much as where you see elsewhere.

Beserk: Action, adventure anime with lot of violence, gore... set in a medieval world. The main character is a mercenary that swings a huge sword. Most of the anime is human vs human fights (castle siege,small-large scale battlefields)... and if enjoyed the anime you should check the manga.

Claymore: Fantasy, action series set in a medieval world with monsters that feed on humans exists and the only ones that can kill them are a group of woman warriors called Claymore. Although it is Human vs Monster fights, there arent "super powers" (fireballs,ninjutsus,magical items,etc). Very interesting plot and story development. There is also the manga that goes beyond the anime plot.

Shura no Toki Martial art, action, historical anime.

I second Claymore and add CANAAN, which has really good action scenes, mostly with guns or knives.

There aren't many, but the fight scenes in Seirei in Moribito are as good as it gets. Lots of budget and beautifully choreographed, like a live-action only better. The one in episode 3 is just off the charts.

Thirding Seirei no Moribito .. The fight in ep 3 is movie quality.. and the other fight wasn't that far behind in quality either. Highly recommended if you looking for a good fight.

+1 . Examurai Senshin.

I add Samurai 7 .. Its basically the anime version of the Akira Kurosawa's Seven Samurai.. however, the fights against the baddies at the end consist of them against giant mechas.. The earlier fights during the collections of the 7 samurais are good tho. I really like the fight between Kanbei vs Kyuzo.