HighwindPilot
White Mage
Actually, I think by technicality, it is an anime. It's the world outside of Japan that refers to their animation as "anime". In its initial creation though, "anime" was no different from "cartoon". Japanese animation is just pretty much what it became, over time, to the rest of the world.
Anyway, your list is very good.
I loved Slayers. Great plot, great action. A definite classic, imo.
Code Geass was also a cool one - I remember it having a lot of suspenseful things about it, including the fights - for mechas though, I think my personal favorite will always be Gundam Wing.
Bakuretsu Tenshi took adjusting for me, but ultimately, I thought it was a solid anime as well.
Outlaw Star was really cool. Outside of like the ending, I thought the series was a great one. Great characters, great dynamics, great variety in "fights" and "style of fights".
s-Cry-Ed was crazy. A really cool show, that had a lot of interesting concepts about it.
And like everyone said, Bebop is well recommended. For me, it was an amazing series. It was the one show that even "non-anime" fans enjoyed. There were a lot of "realistic" yet futuristic elements about it. The characters were interesting, to say the least. It had a great plot going between the present and future. Always an interesting, and sometimes down-right brutal show.
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There are a lot more, but I can't really think right now.
One show that just popped in mind though, is Flame of Recca. Old show, but it was really great. A lot of different elements. A bit like Yu Yu Hakusho in some aspects (to me anyway), with the central Team and everything. The fighting was awesome in it. Check it out.