A list of series and movies I think every anime fan should see.
Arakawa Under the Bridge
Akira
Ghost in the Shell (Original Movie and SAC)
Cowboy Bebop
Fullmetal Alchemist (original or Brotherhood, they’re both good in their own way)
Welcome to the NHK
Kino’s Journey
Mahou Shoujo Madoka Magica
The Melancholy of Suzumiya Haruhi
Anything by Studio Ghibli
Anything by Makoto Shinkai
Anything by Satoshi Kon (but Perfect Blue in particular)
Mind Game
Bakemonogatari
Clannad (After Story is more than worth sitting through the first series)
The Tatami Galaxy
Rurouni Kenshin (the original series and the prequel OVAs)
Haibane Renmei
Mawaru Penguindrum
Revolutionary Girl Utena
Death Note
Kaiba
Tengen Toppa Gurren Lagann
Eden of the East
Sayonara Zetsubou Sensei
Cat Soup
La Maison en Petits Cubes
Time of Eve
Detroit Metal City
Pale Cocoon
UN-GO
Monster
Hellsing Ultimate
The Flowers of Evil
The Eccentric Family
Aoi Bungaku
Natsuyuki Rendezvous
Kids on the Slope
Tamala 2010: A Punk Cat in Space
I will add things to this list as I find them.
Under the Bridge 
I’m curious. Are these anime titles ones that you recommend based on their influence over the medium or are they ones that you found personally moving, or a bit of both? A lot of these titles are ones that don’t affect me viscerally but I recognise their artistry and merit and would recommend that all anime fans try watching them. Nice list, anywho.
Actually that’s a great question. Bit of both, basically. More so than having an influence, I think they’re artistically significant, whether or not this was noticed and/or influenced anyone else.
As a rule, these are all series and movies I personally enjoyed, though there are a few exceptions. Most notably Akira, and Ghost in the Shell, which I could never really get into (original movie or SAC), and are on the list out of impersonal respect. Also, perhaps surprisingly, Studio Ghibli, which I occasionally enjoy but mostly find too childish to get into. I do appreciate their fantastically detailed animation and commitment to strong female protagonists and the like, just the stories often feel childish and shallow to me.
Actually, I should put FLCL up there, too, which I similarly admire but couldn’t get into.
I loved the hell out of everything else on the list though. My personal enjoyment and attempts at more objective critique generally don’t differ much.
Thanks for the comment, bro.