I just finished Neon Genesis: Evangelion last month and was completely blown away by it! The intro alone is by far one of the most iconic in the genre.
Another anime series that asks big questions is Serial Experiments Lain.
It's about a young girl who gets into computers and learns about a parallel world of networked communication called the Wired. The Wired is like VR/AR/internet/chat rooms all in one.
It's only 13 episodes, so it's not a huge time investment, but really interesting. It covers computer hackers, Vannevar Bush's Memex, and more. It's also a bit like a David Lynch story, as it has its own dream logic that doesn't always correspond to what you expect.
I'll second this one. Most of the people in my office who watch anime have all said that SEL played a part in their path in life. It's fun to watch, it has some great artwork, and it's still surprisingly relevant. Just don't expect it to make sense for a long time.
These are all great suggestions but I think it would be helpful to give a small intro to some of them as they are not quite what they seem on the surface. For example, "Kill La Kill" is mostly about exposing the absurdity of fan service in Anime. "One Punch Man" shows that being the best at something is actually the worst thing ever. And "FLCL" is really weird - I truly don't know what it's trying to say.
Demon Slayer just finished last month. Oh, and I missed Attack on Titan and Steins;Gate. Aside from that, I can't think of any other great ones off the top of my head.
Really neat to see this!