Excel Saga, Abenobashi, FLCL, RuroKen, Yu Yu Hakusho, Azumanga, Fullmetal, Hellsing, Lodoss, GITS, and Paranoia Agent (GASP) have already been mentioned, so I'll merely note them as series I enjoyed.
Kino's Journey is one for a bit more of a slower-paced experience. If you like an odyssey story with some deeper themes and more involved stories, this one may be for you.
Dual is worth watching for a slightly different take on the giant robot/mecha theme. A highly overlooked but still quite enjoyable series with a twist on the "boy sees alternate universe and enters it" story.
Tenchi Muyo and its many offspring are a good all-around comedy/love triangle/pentagon/etc. If you like Love Hina and its idea of one boy surrounded by/secretly and not so secretly loved by a bunch of girls, 'tis good.
Seen/liked Voltron/Gatchaman (G-Force and Battle of the Planets)? Try Gravion. Another "Giant robot made from many smaller parts" all around battle. It will remind you of older shows.
A recently released (Not completely released yet) hip-hop infused, modern culture meets ancient times samurai sword swinging fest is Samurai Champloo. A modern twist on such shows as Rurouni Kenshin and Ninja Scroll (Another excellent choice, by the way.).
Classic and modern mecha and giant robots your game? Need an angsty series to complement with Evangelion (Arguably one of the most angsty series of all time.)? Try one of more of various Gundam series. This is the big daddy of the modern giant robot phenomenon (Gigantor started it all back in the '50s). A war of ideals between sides with frequent relationships across sides as well as some of the greatest space mecha battles ever created, Gundam is a superb choice. A similar (But quite different in its own way) series is one of the most classic and famous series, Macross/Robotech.
Want something even more insane than Excel Saga, FLCL, and Magical Shopping Arcade Abenobashi? Try the child of the guys behind the Excel Saga anime, Puni Puni Poemy. WARNING: Not for one who is just getting in to anime. THis really requires some exposure to the genre before it can fully be appreciated, but when you can, a fest of greater insanity is rarely seen.
Vampire Hunter D is one of the best gothic, dark, vampire stories around. Two movies (The original '80s classic and the more recent Bloodlust) showcase the darker side of anime in an impressive way.
Want to see the whole Anime/Manga culture parodied? Try Comic Party, a story, somewhat simply put, about a bunch of Otaku (Anime fans of the highest caliber.) striving to create doujinshi (Fan comics.). An excellent look at one side of the fandom. The soon to be released Genshiken is another example of this type.
Banner/Crest of the Stars is one of the premier (In my humble opinion.) space/sci-fi sagas of the anime world. Battles are present, but what makes this series so merorable are the characters and how they develop. I've rarely seen so much character development is one series before.
Trigun didn't make it so well in Japan, but America loved it. A take on the spaghetti western (Set on a different planet that is almost all desert.), it has battles, comedy, and sadness. Trigun also has some of the more memorable characters in an action series.
Air battleships, clashes in the clouds, and impressive visuals your game? Check out GONZO's magnificent 10th aniversary series, Last Exile. The story of the search for the mysterious Exile, it features fantastic battles between vanships (Fighters of the skies) and massive air battleships, conflict between the more lowly warring empires of Anatoray and Disith and the technologically advanced Guild, it is a most engaging series.
Finally, what list would be complete without the bebop. That's Cowboy Bebop. Some call it the best anime of all time, and nearly all agree it is one of the finer animes around, as evidenced by the fact that CB remains hugely popular years after its original release in both Japan and the rest of the world. CB is the story of a crew of Bounty Hunters aboard the Bebop, a spacecraft, traveling a futuristic solar system chasing bounties and their own pasts. Crazy fights, twists and little additions, and powerful emotion make this more than the ordinary space opera.
I could go on and on, but I really need to shut up now.
And if you buy from Robert's Anime Corner Store, do 'ol HDS a favor and click HDS's referal link before you purchase. You'll earn HDS money, eh?
Clicky!