Blue Submarine No. 6
This short OVA series seems to have picked up popularity in the past couple years after languishing in obscurity since its release. Partially this is because it arrived at an awkward time for anime, when cel animation was shifting to digital and CG wasn't up to the task of bridging that gap, so basically there's a lot of dated computer animation in this one. But it can't obscure the fantastic creature designs and somber environmental story, which becomes more prescient with every passing year. More than lecturing its audience about the dangers of unrestricted growth and looming industrial collapse, it focuses over all things on the sanctity of life, and the crucial importance of coexistence.
Crayon Shin-chan Movies
A lot of people may remember the Shin-chan TV show, which was localized and ran on Western networks during the 2000s. It was a hilarious, occasionally raunchy, squiggly-line comedy about a precocious preschooler's misadventures. And that's fine. But I'm here to tell you that the Shin-chan movies--of which there are over ten, and counting--are on a totally different level. With much more ambitious stories and animation budgets, they're still hilarious and irreverent, yes, but also heartfelt mediations on childhood, parenthood, growing up middle-class, and enjoying the little moments in life. They deserve to be watched as a child and rewatched as an adult or parent.
Dragon Ball
So I could never get into DBZ, despite being a kid during the absolute height of its popularity. I couldn't stand the endlessly dragged-on battles, the obsession with power-level comparisons, and the shameless recycling of animations. That's why discovering that the original Dragon Ball series is notably different was so delightful. All the "useless" characters who mostly stand around and commentate Goku's fights in DBZ have actual roles here, the tone is much more lighthearted and adventurous, and Toriyama's utterly unique world--a place where space tech and fantasy martial arts coexist--is not yet dominated by multi-dozen episode fighting tournaments. An absolute treat.
Future Boy Conan
There are a lot of Miyazaki productions I could put here, from Kiki's Delivery Service to Spirited Away to Heidi, but Future Boy Conan holds a special place in my heart. This is the first full-bodied realization of Miyazaki's two conflicting artistic obsessions: anti-war messages and grand, dazzling displays of human engineering. The world of Conan is a land of endless adventure and unbridled imagination down to every detail, and it's such an embarrassment of riches in terms of animation techniques, environmental/mechanical designs, and direction. This is like the Platonic ideal of the adventure serial you'd be obsessed with as a child, then fondly return to throughout your adulthood.
Giant Robo: The Day the Earth Stood Still
It's very hard to explain this OVA series, even though it probably has the most immediate appeal of the all the animes listed here. It's basically a sci-fantasy crossover miniseries featuring characters from various works by Mitsuteru Yokoyama, including star detectives, medieval Chinese martial artists, mad scientists, spell-casting monks, and--of course--giant robots. It's a dazzling action spectacle with what are hands-down the best action sequences I've ever seen in anime, from a time when studios could have the luxury of spending upwards of a year working on the next episode of a series. And it even has something to say about our world, even if it fumbles the message a bit.
Haibane Renmei
The late nineties/early 2000s saw a minor explosion of taciturn, philosophically-minded anime series like Lain, Boogiepop Phantom, Kino's Journey, the YKK OVA, .hack//SIGN and others, but none have been as special to me as Haibane Renmei. An ambigious slice-of-life fantasy about a group of angel-winged girls living in a sleepy, idyllic village (are they dead? In Purgatory?), it's a gently heartrending depiction of a world that could be, but isn't. It feels like a story about forgiveness and taking life one day at a time, appreciating the simple things and so on. But it's also an almost-liturgical exploration of what is our purpose on Earth and what should we do with the time we're given.
Kare Kano aka His and Her Circumstances
Kare Kano was given a raw deal. After 16-or-so episodes directed by none other than Hideaki Anno of Evangelion fame, the mangaka stepped in because she didn't like the direction the show was going in. Anno was sacked, the new director ruined the show in record time, and the anime never got a second season or satisfying conclusion. But what remains is still one of the most poignant and down-to-earth shoujo series I've ever seen, while also being riotously funny and chock-full of delightful supporting characters. More than anything else, though, the central couple of Kare Kano stands out, for feeling so shockingly human and real, while also reveling in anime's stylistic excess.
Kemonozume
Masaaki Yuasa is one of the most distinctive auteurs working in anime today, and I really liked his work with Devilman, The Night Is Short, and others. But sometimes I feel like he gets caught up in the trademark "weirdness" of his directorial style when, ultimately, most of his works have very straightforward existentialist messages: love and dreams and worth fighting for, even in the face of reality. Ironically, Kemonozume is one of Yuasa's weirdest, most visually off-putting, and also most depraved productions, but also arguably the most honest, as a rogue's gallery of animators and a ridiculous procession of contrived plot points fail to obscure what is ultimately a very screwed-up love story.
Lupin III TV Specials
I'm kind of cheating, but it's very hard to pick one Lupin series to represent the franchise. Part 2 is a classic of serialized cartoons, Part 5 was a welcome return to form, The Woman Called Fujiko Mine is one of the most aesthetically rich anime series I've ever seen... to say nothing of the movies like Castle of Cagliostro. But I think the yearly TV specials, which have aired in Japan since the late eighties, are the most well-rounded representation of Lupin the gentleman thief, the hardened international criminal, the Bugs Bunny expy, the comical lothario, etc. Just a series of consistently great action capers brimming with imagination, and memorable characters.
Mobile Suit Victory Gundam
Thanks to the Gundam franchise's massive, inter-generational popularity, one can sometimes lose sight of the fact that it was intended as an anti-war series from the beginning, dramatizing the meaningless horrors of armed conflict with a shiny new coat of paint for the sci-fi age. In that sense I think Victory Gundam is the most successful entry in the series: it's certainly tragic, though not as insanely oppressive as something like Space Runaway Ideon. It is not the most consistent Gundam series out there or even necessarily the best, and it's not a good place for people looking to get into the franchise either, but I found it to be the most resonant of the ones I've seen.
Patlabor 1 + 2 Movies
Once again I am cheating, but the first two Patlabor movies stand so tall above the TV adaptation that it's impossible not to give them the nod, and they have a very episodic quality about them anyway. While the TV series is a fun workplace comedy/buddy cop show, the first movie is an incredible mecha conspiracy thriller with a breathtaking third act, and the second movie is hands-down one of the best anime productions ever, the full realization of Mamoru Oshii's directorial ambitions. A dark, glacial political thriller about the ways in which countries wage war in the 21st century, the futility of trying to change systems from within, and the way in which nations bury the victims of their history.
Perfect Blue
It's no wonder this movie served as such direct inspiration for the likes of Black Swan; it sometimes doesn't feel like anime at all. It uses so many cinematic techniques that are normally found in live action, and the acting is so naturalistic, that it can fool you at times, even though your eyes are consciously registering the stylized designs and cel-based animation. Perfect Blue is just a really solid and endlessly rewatchable psychological thriller, and also a chilling look into otaku/idol/celebrity culture, the Web 1.0 rumor mill, the callousness of the entertainment industry, and the ways in which we fragment our own identities to survive the trauma of our lives. Satoshi Kon's masterwork, IMO.
Polar Bear Cafe
There are TV shows that function as spiritual comfort food; for some people it's their favorite sitcom, or a heavily-serialized childhood cartoon. For me it's Polar Bear Cafe. I wish there was a TV channel that aired Polar Bear Cafe episodes 24/7. It's a placid, good-natured slice-of-life series about a world in which humans coexist with sentient animals who live, talk, and work alongside them. It's a deeply charming show that very slowly unveils hints of world-building and character back stories, but don't expect any "lore" or dramatic twists out of it. It's something nice to watch with coffee in hand, or unwinding at the end of the day. It is, probably, what a perfect world would look like.
Revolutionary Girl Utena
There are two anime series which, if encountered at the right time in your personal development, can proceed to shape your identity and outlook on life for years to come: the first is Evangelion (to the detriment of many), and the second is Utena. While Evangelion portrays the need for human connection as a grand struggle between giant robots and space aliens, Utena examines the ways in which the people in our lives play roles and live out elaborate fantasies, which--intentionally or otherwise--can end up being just as destructive. Both series are about teenagers and the end of the world, but Utena arrives at a much more resonant message while also being aesthetically superior.
Tekkonkinkreet
This movie is an outlier in many ways: a magical-realist adventure about a pair of street-urchin brothers who rule over a bustling, hyper-colorful metropolis, directed by a Westerner? And based on a manga that's about as far from the mainstream as possible, yet provided with a generous budget? I'm still not sure how it happened, but I'm glad it did. Tekkonkinkreet is a celebration of brotherhood, local color, street culture, youthful exhuberance, the messy excitement of big cities, and the culture-clash qualities of contemporary Japan. It's also a touching story about forgiveness, self-reliance, and unconditional love. So it has something for everyone.
Runner-ups:
Cat Soup
Descending Stories: Shouwa Genroku Rakugo Shinjuu
Kekkai Sensen aka Blood Blockade Battlefront
Kyoukaisenjou no Horizon
Lucky Star
Natsume Yuujinchou aka Natsume's Book of Friends
Night on the Galactic Railroad
Royal Space Force: The Wings of Honneamise
Turn A Gundam