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I've kept this list to series I've finished, since then I don't need to worry about a series going bad on me. The ordering is also subject to change depending on my mood. Also, this is JUST THE TV SERIES, any related movies are being ignored in this list unless stated otherwise.

1.) Fullmetal Alchemist - This series, in my opinion, pulled off everything extremely well. The plot and pacing were good, the characters were well-developed and didn't fit into standard 'good' and 'evil', the story was pretty much a perfect length... overall I can point out very little that I disliked about the series. Al still is one of, if not my, favorite character of all time.
2.) Kino's Journey - I just recently finished the series, but I can guarantee that it's very deserving of this spot. The odds are good that you haven't seen a series like this, and there is a charm in discovering such strangely beautiful places in this series.
3.) Cardcaptor Sakura - Even if the pairings are pulled off better in the manga, I love the anime. The characters are so well-developed that I can't find a reason to even disike any of them, and the cards and costume designs all look really nice, even the ones created for the anime. It's also nice to see a magical girl series made for girls, since the genre been geared towards boys more recently.
4.) Paranoia Agent - There are few things more interesting than watching characters slowly reach the breaking point, only to be attacked by a kid with a baseball bat. The plotting of the series is also really interesting because it will focus on a different character most episodes, but it never feels episodic or repetitive.
5.) AzuManga Daioh - One of the deepest stories I've ever found. An absolutely hilarious series, and odds are you'll relate to at least one of the girls.
6.) Neon Genesis Evangelion - There's always something weird about watching a series and finding a character that reminds you of yourself so much it's eerie. I'm also not a mecha person, so I enjoyed that the series has mecha aspects, but doesn't focus on the battles.
7.) Serial Experiments Lain - I'm going to have to rewatch this series later just to see if I can get any more from it. It's a great series, but it won't make sense your first time through.
8.) The Melancholy of Haruhi Suzumiya - Odds are, any reasons I can give for enjoying the series have been said far too many times to count.
9.) Air TV - Someone described Air as being a 'fairytale for adults'. I have to agree. The story is great and the series has a very nice feel to it, despite how good or bad things go. The characters are also very well-developed.
10.) Ghost in the Shell (the movie) - Unfortunately, this movie's too short for me to really explain WHY I like it. sweatdrop

Honorable Mention (a.k.a. Shows that are good, but I haven't finished):
Death Note
Higurashi no Naku Koroni / When They Cry
Kashimashi ~Girl Meets Girl~
Kanon (2006)
Lucky Star
Sayonara Zetsubou Sensei

As you can probably tell, I like deeper anime. I love comedies as much as the next guy, but I'm not so much for the action shows as I am for the shows that bring up interesting ideas and questions, or the psychological dramas.