Haven't really watched many series and lost interest sometime around when I grew up.
No but seriously I lost interest in anime, and I became more manga-oriented (not a huge manga otaku though) but there are still some anime I enjoy a whole lot, all of which have been recommended already.
Death Note and Code Geass obviously come to mind, two of the best series with great plot depth, it's pretty good for anyone who likes good plots with some smart twists. I somehow like the anime of DN better than the manga.
Giant robots is always cool, especially for people who are really into that kind of stuff. Gurren Lagann would be a good start, pretty absurd but pretty awesome. Then there's Gundam, the original series was pretty good though a little bland, Zeta is personally my favourite, but it'd make more sense if you watch the original series first.
And of course Spirited Away.
I personally don't see a the difference between an anime and a normal TV series or a movie, it's just a different graphic style, the plot and the concept are the most important underlying factors, so I would just recommend someone a good anime like I would recommend them a good movie.
If you people consider cartoons as anime, then Futurama, it beats most anime or TV series, funny and smart. As hell.