This has to be one of the best anime I have ever seen, though sadly there are only 6 episodes.
Naota's life is confined to going to school and living with his father and grandfather. The usually tranquil life in Mabase is rudely interrupted by the arrival of Haruhara Haruko, who bursts on the scene by running Naota over with her Vespa scooter and hitting him on the head with a Rickenbacker bass guitar. Later, Naota is shocked to find Haruko working in his house as a live-in maid.
Haruko's search for the alien being Atomsk puts her at odds with Medical Mechanica. At the same time, Naota is being watched by Commander Amarao. The Commander believes Haruko is in love with Atomsk and Medical Mechanica is out to conquer the galaxy. The fortuitous circumstances get Naota involved in a three way battle between Haruko, Amarao and Medical Mechanica.
FLCL's odd style, hyperactive pace, convoluted, esoteric plot, and tendency to break the fourth wall sets it apart from other contemporary anime. It can be categorized as a work of comedy, drama, science fiction, romance, philosophy, poetry, and a parody of contemporary culture in general. It is an abstract designed storyline about growing up and losing the childish viewpoint of life, that all people once had, and seeing the true reality of the world.
(taken from Wikipedia)
This is even one of those Animes which the dubbed version is as good as the subbed.