@lackingsaint: Gurren Lagann is one of my all-time favorites, but I wouldn't recommend it to a newcomer. I feel like it loses a lot of its impact if you have no context for the roles that giant robots tend to play in anime. Psycho-Pass, though... that's one that I've recommended to newcomers before. I actually love Psycho-Pass for some very specific reasons, but we don't need to get into that here~
In general, I agree with what @golguin is saying in that last post - there are certain shows that have proven to be popular with a general audience, particularly people who haven't seen much anime before. Attack on Titan isn't the cream of the anime crop (then again, nobody really agrees on what is in the first place), but it's DEFINITELY entertaining and interesting to a huge number of people. It also doesn't require any knowledge of foreign culture or particular types of humor or concepts like "moe" or anything along those lines to enjoy. This is probably its biggest selling point, in my opinion - and also a big part of why I think shows like FMA or Cowboy Bebop fulfilled a similar role a decade ago on Toonami and Adult Swim.