Serial Experiments Lain is an anime about a young girl named Lain. The series is about 13 episodes each half an hour in length. It is available in both subbed (subtitled) and dubbed (japanese voices redone by english-speaking actors) formats and is a very intelligent look at god, the collective unconscious, the internet, and many other of those themes most common in cyberpunk literature. As a note, both the subbed and dubbed versions have their pros and cons. The translations were for the most part better (IMHO) in the sub. However, one character's name is translated as Arisu, which is the Japanese way of saying Alice. In the dubbed version however her name is Alice. This is important, as it was meant to be a reference to Lewis Carrol's Alice in Wonderland.
This series is an excellent example of how anime can require a lot of thinking to enjoy. The whole series is very confusing at the beginning and raises two new questions for each one it answers. At the end you're left with several possible explanations to choose from, or it is very easy to come up with one's own explanation.