Mecha Madness
The anime that invented 'Real Robot', though in retrospect it got pretty goofy too.
Mobile Suit Gundam
Bubblegum Crisis 2032
Sailor Moon’s famous line.
“In the name of the moon, I’ll punish you!”
"I ask of you..."
"Are you my master?"
The very last line of End of Evangelion, in English or Japanese.
"How Disgusting."
"Kimochi Waru."
It invented shonen battle franchises. Make no mistake about that. You know what it is.
Dragon Ball (Z)
Written by the guy who wrote Hunter x Hunter.
Yu Yu Hakusho
He's a dog demon with a sword. She's a time-travelling high school girl. They’re about to fall in love.
Inuyasha.
A bishonen young lad discovers he has mind control powers and tries to do something about the British. Pizza Hut is involved.
Code Geass
The psychic child that levelled Tokyo in 1988 - and was dissected by the government to unlock the secrets of his apocalyptic power.
Akira
Stars a martial arts himbo who worries about very little despite multiple bloodthirsty rivals, multiple fiancees, and an ancient Chinese curse in his life.
Ranma 1/2
The One With the Gag Dub.
Ghost Stories
One of CLAMP's earlier works, it's an isekai (before there was such a word) about three high school girls with big swords and a bigger mech.
Magic Knight Rayearth
Spiritual predecessor to Princess Mononoke, one of Miyazaki's first really big films, about the post apocalypse where a toxic forest threatens to swallow the earth (or does it?)
Nausicaa of the valley of the wind
Cowboy Bebop, Episode 13:
Gren: "I believed in you."
Vicious: "_______"
"There was nothing to believe."
We watched this in SF week last quarter, an adaptation of an incredibly influential manga focusing on a squad of elite police cyborgs in a cyberpunk future. Brain transplants into totally artificial bodies, and hacking those brains in turn, are common plot elements.
Ghost in the Shell: Stand Alone Complex
Masamune Shirow's OTHER big cops n' robbers show involving mecha-equipped police. Cyborg catgirls are involved.
Dominion Tank Police
Mamoru Oshii made this police-oriented mecha show towards the end of the 80's, and went on to make several more anime adaptations.
Mobile Police Patlabor
Similar to Cowboy Bebop, this anime stars a lone wanderer whose bad reputation causes more trouble than he does.
Trigun
Moe, moe, moe, moe! It's got to be moe, otherwise it isn't exciting!
The Melancholy of Haruhi Suzumiya