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My recommendations:
Cannot agree any more with Cyberpunk egderunners. It soared into my top 10 anime list. I still tear up when listening to that song
I mean I know I gotta watch Edgerunners as I am really into the Cyberpunk lore and even play Cyberpunk Red, the ttrpg so yea π
Yes I loved Edgerunners because it is so short so not a huge time investment to get into and wrapped up in one shot. And I didn't play the game, but the anime made me really want to play it. I am however waiting for the DLC to come out before venturing into the game.
This music video is worth a watch too which shows a character from before the timeline of Edgerunners. Can watch before or after.
Seconding Akira and Ghost in the shell, Steins;Gate is great too.
One of the first ones I saw (and I love cyberpunk and scifi shit) was Serial Experiments Lain and it was one of the first ones to get me absolutely hooked. Mind you, it is weird and strange, but if you're into stuff like David Lynch at all you'll love it.
Others to look into that aren't exactly cyberpunk, but futuristic/scifi nonetheless: Neon Genesis Evangelion (finish it, it gets really cool towards the end and I thought all things mecha-anything was just dudes being silly about robots), Cowboy Bebop, Ergo Proxy and Gantz (might seem silly at first, but gets interesting towards the end).
I have seen Lain, and loved it so much
Gotta start with the classics - Akira is the universal Anime film that others get judged by. Ghost in the Shell, as ModularTable said. Bubblegum Crisis Battle Angel Alita
Steins;gate doesnβt exactly fit your criteria, but I think itβs worth a watch anyway. Itβs a really quality time travel story.
I love Steins;Gate! I will warn - it has a slow burner start but it really is worth it.
I feel like the slow burn really adds to it. Iβd treat it like a slice of life anime for the first half.
I wish I hadn't ruined my first watch of Steins;Gate by starting from one of the last episodes. It got me excited, but there was so much stuff referring to earlier episodes that made me realize I'm not watching the first episode. I'm waiting just a bit longer to restart watching. I did the same with Battlestar Galactica's last season and got all excited like "Holy shit this last season is going to be crazy" before realizing I just completely ruined the whole season for myself.
Things that come to mind quickly are Ghost in the Shell, Ergo Proxy, Texhnolyze, Cyberpunk Edgerunners, Psycho-pass.
Serial Experiments Lain isn't really cyberpunk but has aged to be retrofuturism.
I'll echo the recommendation of Neon Genesis Evangelion, particularly if you're looking for a story with insight into the psychological condition and experiences of the protagonist.
If you're in the mood to ponder other ideas philosophically, you may want to try Time of Eve (Eve no Jikan), Plastic Memories, and Beatless. All of them are stories involving androids, but they explore different concepts. Time of Eve explores issues of identity and presentation (in some ways that are surprisingly salient today), Plastic Memories explores end-of-life considerations, and Beatless explores the roles of AI in the future of human society.
For a couple series with dystopian futures, consider Coppelion and Planetarian. Coppelion is about people who venture into the highly radioactive ruins of Tokyo after a nuclear accident; its visual style deliberately makes use of washed-out colors. Planetarian is about a scrap collector who ventures into a ruined city only to encounter an android who seems oblivious of the ruins around her; her wordy and oblivious optimism contrasts sharply with his cynicism, in this relatively short and also somewhat philosophical anime series. (It's also available in the form of a visual novel.)
For something more action-packed, there's Guilty Crown and the Symphogear series. Guilty Crown seems to have some polarizing opinions, though not for lack of effort in the production -- it's quite a spectacle, visually and musically, and it was definitely meant to go big. On one hand I felt disappointed by it but on the other hand I also feel like I really wanted to like it. As with any show, though, you should be forming your opinion of things anyway.
For drama, I'd suggest The Price of Smiles (Egao no Daika) and Str.A.In: Strategic Armored Infantry (Soukou no Strain). Both of them revolve around war and associated tragedies, but in settings with futuristic combat mechs and other technological advancements. The latter takes place in space.
For something with a brighter but still futuristic vibe, try Stellvia of the Universe (Uchuu no Stellvia). Here's a story about high school in space, with all the laughs, goofs, friendship, smiles, and even romance, as well as interpersonal drama and expectations -- particularly heavy expectations at that. As a point of trivia, it's pretty neat that the main character specializes in programming.
Edit: I wanted to throw in one more recommendation, the Nanoha series. This is a magical girl series that's renowned for incorporating Gundam-inspired magitech elements, particularly in the various magical abilities the characters use. It's also got both drama and positive vibes. Good for an affirmation of the power of friendship, backed up by a big stick!
The anime series Ghost in the Shell and Cowboy Bepop both fit that description in their own way.
I think you'll also like the anthologies: Neo Tokyo and Memories.
And the movies: Redline, Akira, Metropolis and the original Ghost in the Shell.
I also need to mention Nausicca of the Valley of the Wind, Laputa: Castle in the Sky and Summer Wars. These ones aren't exactly Cyberpunk but I would say they scratch the same itch; only in a different way.
*I also suggest you read the manga BLAME!, it's about as cyberpunk as you can get. Just be sure to read happy, regular things at the same time so you don't get depressed.
It's old but my absolute favorite sci-fi anime is Gunbuster/Top o Nerae! (Aim for the Top! in English).