This question and the answers have been quite confusing...
"These are the anime that when you mention them to others who claim they are fans of anime and yet if they don't know what you are talking about you look at them like do you live in a cave?" This really makes me drop out all the Narutos and DBZs and Pokemons from the list since god dammit, even the animehaters know these. And my personal opinion (no offense), if those are the ONLY anime you know and you call yourself an animefan while snorting at everything else that isn't Naruto, OP, Bleach or something similar, you don't even have the right to call yourself an animefan... :P
Now, for the list...first one that pops onto my mind is Neon Genesis Evangelion, which is both hated and loved by many and I'm the guy who has mixed feelings and likes the manga better... xD. Anyways it's still a great anime whatwith the last two episodes being done with no money and therefore being kinda...dissapointing ending (although the EoE patched the ending quite nicely...).
Then there's Cowboy Bebop. I don't really know much about this, I've watched 3 episodes so far, nothing amazing yet....I'll keep watching them every once in a while. Although truth to be told, I prefer an anime with a clear beginning and ending (like Steins;Gate or Anohana) and the episodic feeling isn't really for me... :P. Now that I started this ranting about storylines, I might as well progress with my "Why Naruto, OP, Bleach, DBZ and such aren't that great as they're made out to be" -rant: the neverending story feel that I get from those series (especially Bleach lately) gives me the impression that the author is just milking the series and going with a predictable pattern on and on until even the slowest of us realize that he's just making **** up as he goes and the plot, the uninteresting generic characters and the world are just a big pile of ****.
Oops, it seems I strayed from the subject a bit. But to make it simple, Cowboy Bebop isn't really my cup of tea but it really is one of those "classics" that the animefans should really at least know.
And onto the next one: I think I'll go with Code Geass. It's a lot like Death Note and it's in my list mostly because DN wasn't originally an anime, so CG can take it's place...there's not enough room for both of them and due to their similarity the one anime that is "more of a pure anime" gets to stay. All in all, really intense, what with some needless stuff on the second season and such...Overall really entertaining with a great ending.
Welcome to the NHK! ...This may very well be on the list only because I just watched it but damn, that was probably the best, most realistic and definitely one of the few animes that everyone can relate to at least a little...(unless you're some douchebag with no thoughts or emotions...) Lots of dark humor, modern psychological issues and dealing with them, getting secluded from the society, internet suicides, conspiracy theories....this anime has it all. Don't let the hentaigame stuff fool you...it really gets deep when the plot thickens...(Well, "some kind of a plot", it's a slice of life but whatever, it still has some!)
What we're on number 4 already? Hmm....those hentai games remind me of the final candidate. Actually it kinda fits the theme perfectly, a show about otakus: Genshiken! (Tatta-da-daa!). Yeah, although it has an anime, definitely read the manga, since it goes further... Really enjoyable, all otakus should be able to at least somewhat relate to the characters... :D
Hmm...let's see that list again...
1) Neon Genesis Evangelion
2) Cowboy Bebop
3) Code Geass
4) Welcome to the NHK!
5) Genshiken
Yeah, I kinda just went with what popped onto my thoughts... I could be missing something huge but whatever... :D