I'm amazed that no one has actually made this recommendation yet.
Top wo Nerae is the Onee-chan of NGE by a good seven years. Both are works by Hideaki Anno, but NGE gets all the glory for being more recent (which is a shame in my opinion).
Top wo Nerae is a truly groundbreaking, character driven, Sci-Fi anime that is better NGE in many respects. If Top wo Nerae did not exist, then NGE may never have been made...
Both are powerful well written mech stories and Both are animated by Gainex,unlike it's later Sequel Diebuster,The first Gunbuster was more seriouse and payed more homage to Evangelion,both the first Gunbuster series and Evangelion both feature a lead who both grow from cowards to strong leads, Noriko Takaya from Gunbuster is very similar to Shinji Hikari from Evangelion,they may be crybabies at times,but they're likable and tolerable,Gunbuster is a cult classic worth checking out if you like Evangelion (Aliens,Angels, and Fanservice.....Oh my) and a bit of horror
Classic Gainax mecha anime that had a huge influence on the genre of mecha anime as we know it today.
Both are mecha-themed series created by Hideaki Anno. Though Gunbuster is seven years older, it already covers the concepts that formed the basis for NGE. Gunbuster does not delve in psychology as much as NGE, yet it tells the same coming-of-age story in which kids have to fight in real battles.
Giant super robots, massive battles against horrific enemies and unpleasable father figures! (And nudity!)
Hideki Anno created both Top wo Nerae! (Gunbuster) and Evangelion. Gunbuster is an essential for any mecha fan and a must watch for Evangelion fans because you can clearly see the prototypes for characters and ideas in Evangelion.
Coach = Gendo, Jung-Frued = Asuka, etc.
If you haven't seen Gunbuster, you can't understand what Anno is deconstrucing in Neon Genesis Evangelion.
You can easily see how NGE and TTGL grew from Top wo Nerae. It's as if a branching pathway was created by exploring the two extremes of psychological fear from piloting a giant robot and the over the top nature of the anime.
The first direction of Anno, Top wo nerae shares a lot of aspects with NGE (themes, characters, direction...), while in a denser package. A must-see for any SF anime fan (and a "watch-it-right-nao" for Anno fans).
Both star an unlikely hero who ends up saving people with a giant robot which the show is named after.
Both also ran out of money.
There are a lot of similarities between these shows. Main characters are similar and they have similar character arcs. Both deal with large robots fighting invaders from space, although NGE is mostly about combat on Earth while Gunbuster focuses on space battles.
Classic Gainax shows that made a major influence on the mecha genre and Gainax itself, with Gunbuster being more accessible and NGE being more psychological.
Both made by Hideaki Anno, mechs vs aliens, huge amount of attention to detail and science, setting up a tight and believable story with human characters and realistic moral conundrums.
Eva is darker/deeper, though Gunbuster has spectacular tits. Take that as you will.
Both are mecha series from Hideaki Anno with emphasis on characterization, drama and storytelling rather than action. It isn't as heady or as allegorical, which might be better for those who think Eva is too pretentious.