In the English Dub, when Asuka first meets Rei, she calls her behavior Freakazoid!.
Too many to various world religions (particularly Judaism and Christianity) to list, or possibly even detect.
For a start, all the Angels are named after Angels from Jewish, Christian and Islamic theology and incorporate subtle parallels with their counterparts' names and functions.
Misato initially drinks "Yebisu" beer, a real brand that is depicted correctly down to the label, followed by "Yebichu" beer, a parody of the real label which references the manga Oruchuban Ebichu, of which both Anno and Japanese voice actress Kotono Mitsuishi were fans. Anno later developed it into an anime at Mitsuishi's recommendation.
The Human Instrumentality Project was named after Cordwainer Smith's Instrumentality of Mankind, though they are not otherwise related.
In turn, the Hedgehog's Dilemma is an allusion to Schopenhauer. It's also been theorized among fans that the original ending alludes to and borrows from some of the existentialists, such as Sartre and Kierkegaard.
Also of note is that working title for the last episode as per the series proposal was "The Only Neat Thing to Do", the same title as a sci-fi short story written by Alice Sheldon under her Moustache de PlumePen Name of James Tiptree Jr.
Toji and Kensuke are allusions to characters from Ryu Murakami's Ai to Gensou no Fascism (The Fascism of Love and Fantasy), from which Anno borrowed much of the psychological material.
Keel Lorenz is an allusion to the ethologist Konrad Lorenz. He was originally named Konrad, too.
Konrad Lorenz developed the concept of imprinting in terms of mother/child relationships. For example, say you've got a duck's egg in an incubator, and it's ready to hatch. The eggshell cracks, and the chick's head pops out. It sees you, and thinks, "Mother!"
Interestingly, this series is the one most referenced by later Gainax works, and possibly anime in general. Yes, even Tengen Toppa Gurren Lagann.
The manga seems to have shout outs to Star Wars, of all things. For example, on a NERV writeup of Toji, there's a documentation of his Midichlorian count. In a recent chapter, Gendo also pulls a Darth Vader on Shinji, which is reminiscent of the fact he loses a hand in The End.
Misato mutters "I have a bad feeling about this" in the episode with Jet Alone.
The style of the opening is based on the classic sci-fi show UFO. Furthermore, the character design for Gendo Ikari and Kozo Fuyutsuki are animated look-alikes to the actors Ed Bishop and George Sewell as characters Ed Straker and Alec Freeman (the commander and second-in-command from S.H.A.D.O., respectively). Also the "blink-and-you'll-miss-them" title cards that appear at times are taken from the show.
The title of the latter half of The End of Evangelion, "Magokoro wo, Kimi ni," comes from the Japanese title of the movie Charly, which was based on the short story/novel Flowers for Algernon.
Devilman and the Ultra Series get multiple shout outs throughout the series, some subtle, others overt.
According to Anno the Devilman ones were unintentional and he only noticed the similarities when they where pointed out to him. Interestingly, one of the two unused, proposed endings to Episode 26/End of Evangelion was even closer to Devilman's ending than the final version.
A blink-and-you'll-miss-it one comes in form of one of Shinji's t-shirts, which has the logo of the British New Wave Music band XTC on the back. Seeing as quite a few of XTC's songs revolves around some of the same themes as Evangelion, the reference is likely more than just a superficial one.
The scene where the military attacks NERV HQ is an homage to Hideaki Anno's favorite film, Battle of Okinawa, right down to the use of flamethrowers.
There are multiple references to Michael Crichton's The Andromeda Strain, mostly the film version (the use of the "601" graphic to showcase the MAGI having a computer analysis error, the design of the Geofront resembles the Wildfire complex when seen from the side, and most importantly the episode "The Lilliputian Hitcher" is almost-but-not-quite a Whole Plot Reference).
Shinji’s and Asuka’s Masochism Tango is disturbingly similar to that of George and Martha from Who's Afraid of Virginia Woolf?: Asuka is disgusted by Shinji’s lack of assertion, Martha is frustrated with George’s, attempts by either at appeasement only aggravates them further, and both of their frustrations have to do with sexual performance and assertion—Shinji won’t cross the ‘Wall of Jericho’ and notoriously settles for masturbating by her bed when she’s comatose while George is implied to be unable to perform, and both are Ambiguously Gay. Both Shinji and George ultimately both snap and lash out destructively, complete with an attempt to strangle Asuka/Martha.