Ahh a subject I can enjoy talking at length about!
#1 - Cowboy Bebop: A masterpiece of character story telling, manging to not have a central throughline in terms of plot without feeling crazy. Simple, relatable and interesting characters, coupled with great music, great fights, and interesting and believeable world and knew when its arc was done and ended it. The intra-character relationships are extremely interesting without feeling arty, existing just as what they are unapologetically within the odd world they inhabit.
#2 - Full Metal Alchemist: Brotherhood: Like bebob, it exactly spends the time it needs to tell the story it is telling without trying hard to inflate its events or run forever (Read DBZ / One Piece / Naruto, etc which are built around a consistant cast but without an end in mind). Interesting characters, designs and the, though perhaps overt, use of physical combat for characters fighting with their interal sins (only allowing these fights to be successful as that character overcomes that flaw, Scar beating Wrath only after he has put aside his hatred and blind desire for revenge, that everyone generally fails to defeat Pride aside those who have struggled with that emotion and overcome it emotionally).
#3 - Jojo's Bizzare Adventure: Man it is just a trip. It's odd and knows it is odd, but delivers cool characters super high visual style and a neat narrative that sorta just does its own thing, exisiting as a unique identity that's hard to immitate. The use of color and shape is so well done that it excuses the totally insane plot and characters that exist, though season 1 has some 'this recurring character is a super huge litteral nazi, and one is an italian supporter of nazism and the show sorta fails to be aware of this' which does diminish that 2nd arc, but is largely the only critizim I have of it.
#4 - Ghost in the Shell: Stand Alone Complex: While both the original movie (and its sequal), and 2nd Gig have their own place (Along with its Manga heritage), in particular Stand Alone Complex I love for its interesting character arcs, awesome world and tech, while talking philosophy as said tech advances creating new problems all the while an interesting crime drama happens during an investigation.
#5 - Trigun: An older one with a lower budget than the other three, a sort of anime doctor Who, the timeless godlike being who wanders with ulteristic motives. An amazing style, a great cast, etc. It's a bit prechy with its protaganist's hard 'no kill policy', but often goes out of its way to show the harm that concept can cause to both the person holding the belief and those around him, while also showing the importance these concepts have.
#6 - Mobile Suit Gundam: Iron Blooded Orphans: I was going to limit this to an honorable mention below, but it just fails to do it service. I think this show is great, and can stand entirely by itself without other Gundam entries. Like Trigun it allows the normal anime protag theme of 'trying really hard and overcoming through force of will' but taxes its heroes with horrible personal costs at every turn (The protag in this case becomes a life time cripple, first losing use of an arm, and latter, use of his legs). And it manages to do this without wallowing in pity. It has -great- fights, awesome drama and a great central story and theme.
#7 - Mushi-Ishi: Fuck I really ment to do only 5 of these, but MAN Mushi-Ishi is truly legendary. An episodic telling of a wandering spirit sooth sayer solving unrelated problems of spirits turned malicious or otherwise causing problems. It is a slow paced show orbiting Gecko (said sooth sayer) with an amazing visual style.
#8 - Now and Then, Here and There: Well I'm over 5, why not do ten? This single season show follows the impossibly determined child through a trip into an alternate reality. I've never had a show better show a characters personal convictions, and the reverbations it create in the world around them in such a pronounced way.
#9 - Boogiepop: Phantom: Immensely dense and complex, positing social and temporal realities which overlap and interact. Based on a series of novels and manga, it uses ultra dense symbolism and metaphor and super not for everyone. But I do rate it as a show that's likely had the most impact on how I see the world. Told as a pulp fiction, each episode following a different character, each orbiting a familiar cast, parting out the story from different persectives, moments in it's progression, all to illistrate a point about how history and soical interaction is shaped by personal perspectives. A show that asks the viewer to pay attention and piece things together, expect to have to watch it 2 or 3 times to really get all the information needed to draw real conclusions. Though I liked it a touch less, the new season is also great, four short stories making 3-4 eposiode arcs following a particular character (rather than shifting every episode), and much more 'about' the shows two core characters (Nagi Kirma and Boogiepop), though the latter very indirectly (as the character is weird and the show itself generally makes it unclear exactly what Boogiepop is (a suppressed personality of a character acting rather like a super hero, a supernational manifestation possessing that person for unclear reasons, etc).
#10 - Neon Gensis: Evangelion: The immensely impactful Evangelion, the mech show that posited the concept of "We can have robots punch monsters but also have it be about really depressed children!" It, like Boogiepop above, is at time up its own ass with its symbolic nature, so if you don't like a show more about drama and thinking and at times having to rewatch and be like "The fuck is happening?", maybe look elsewhere, and also look else where if you want a happy or even sensible ending, because this shits gonna get wieeeeeeerd. Still, the effort to detail (like seriously, the first fight in the show of the giant robot spends as much time showing the immense logistical feat that is moving these robots through a city, the crews getting rail systems, power cords, and what not into place. That these robots are not a single pilot saving the day, but the effort of hundreds of men and women working towards a shared goal (Even if their efforts still rest on the shoulder of a single pilot who must bear the responsibility of their hopes). Includes a movie that re-ends the show, and three new movies that are both a reboot and a contiunation of the show. Because that's how wierd the show gets.
The honoerable mentions, Code Geass, One Punch, Black Butler (exclusively season 1), Devil is a Part Timer (a darling slice of life that does so much right in action, drama and comady), Welcome to the NHK (a look at anti-social paranoia and the growing hakkiamori rise in Japan), Outlaw Star (A show in which space ships that -litterally- punch each other is a central event).
I'm sure there are more I could list but these are the ones that spring to mind, limiting myself ONLY to anime Shows (thus exlcluding Akira Mononke, Howls, etc). I could obviously write a bit about the honrable mentions if desired as to why I think they are good, and could write an entire disertation on any of those 10 listed in great detail, and would likely use them as a pretense for rewatching them.