I'm going to be a bit awkward and not give a plain list, but rather talk around a bunch of things to start to explore
why they might be worth having or watching. Besides, I'm no good at turning out (ranked) lists.
First up, Akira.
It's probably not the finest animated film ever but it was something of a landmark for both me, and as far as I know for the UK audience in general. This is the one that showed me that there was something going on in Japanese animation beyond 'cartoons for kids'. Those bikes looked cool back in the 1990s too. And yeah, I am going to mention it. "TETSUOOOOOOOOOOOOO!" "Kane-daaaaaaaaaaaaaaaaaaaa!" That shouting is now the stuff of legend.
Serial Experiments Lain.
Okay, you might see a bit of a theme emerging here. There's nothing about Lain that says 'cartoon for kids'. It's one of those 'clever' series' that's not pleasant to watch, but is somehow satisfying. I watched it again recently and boy has it aged. You are very aware that you are watching something from an earlier era of animation when you watch it now. That in itself is interesting. Much of the technology (and certainly the game graphics in use now) has moved on in leaps and bounds since Lain was made too, and that is also worth paying attention to. I'm sure most of us still don't have computers that say 'Hello (your name)' out loud when we log on, and most of us still use a mouse and keyboard to interface and never say anything like 'Hello Navi' to our machines. Lain also has that adorable bear suit and bear slippers. Which are almost enough to make me want to find a very big bear suit for myself... ¬_¬
Haibane Renmei
You might see another theme emerging with this one. It's Yoshitoshi Abe again. Haven't watched it for a long time, but it's another series which is both 'clever' and trying to say something about life. It may well have been the first anime series that made me cry. If a 'cartoon' can move a man to tears then there must be something to it.
Eden of The East
It's social commentary. It's smart. It's somewhat convoluted but it does come together in a satisfying way for the ending.
Urotsukidoji
Yes it is pretty base 'tentacle rape hentai crap' and yes it really is without much merit at all. But it is interesting in a couple of respects. It is one of the first 'big' titles that were used to spark the UK market and in the process gave 'manga'/anime a very bad name in the press, and it is the first of the anime titles that brought attention to UK censorship and the difference in cultural acceptability of certain themes between the UK and Japan.
Beck: Mongolian Chop Squad
I like music animes. At least I like all the ones I have seen. Now Beck is interesting both for encouraging the audience to have a go at music themselves, and also for the problems with getting some of the music used in the series licenced in territories beyond Japan. I've yet to figure out why they claim that the lead character has a good voice too, but that's another story.
Sakamichi no Apollon (Kids on The Slope)
If you are a Crunchyroll user you will have had the chance to see this already. As mentioned by DIC in an earlier post the rest of you ought to grab it as soon as it comes out. This is one of those top tier animes for quality, style, and musical content. Thoroughly Jazzalicious, very moving at times, and nice to look at. The only slight criticism is it is another of those series' that has a slightly rushed and abrupt ending. Not as rushed as something like Ano Hana mind.
Seto no Hanayome (My Bride Is A Mermaid)
One that has slipped under the radar for a lot of people, but again it's top tier. Top tier comedy this time. Chock full of flying wavy cat jokes and even some fish porn gags! There are also some very sharp 'grown up' remarks and jokes in there too.
Tekkon Kinkreet
It's flawed, and not that great a story but it was visually very impressive when it was released way back around 2006. Perhaps most interesting for being one of the titles that were used to push Blu-ray sales.
Paranoia Agent
It's another of those themes coming out. Satoshi Kon. Yes. It's yet another grown up series. Very impressive, very heartwarming (unless you are a BBFC censor and don't think the expected UK audience will get the gentle humour in a certain scene) and yes I have hugged a Maromi plush.
Barefoot Gen
Quite horrible to watch at times. But exceptionally worthwhile. Not least as at least some indication of how Japan processed the events of WWII through the national psyche.
Grave of The Fireflies
Another one that is horrible to watch. It will probably make you cry too. But who said animation was just to keep kids occupied?
Now and Then, Here and There
Notable for the cruelty and horrible things which happen within the story. Once more, it's a reminder that Japan will show things which you would not expect in a regular cartoon for kids, and that the Japanese are more than capable of doing so in a responsible and high quality way when they want to.
Okay, that's twelve items. There are obviously loads more I could mention but I think that lot might be enough to spark some comments.