And now we get to a personal favourite.
Urusei Yatsura is the show that was the first subtitled anime I saw, and the show that sealed my interest in anime.
For those not familiar with Rumiko Takahashi's story, it involves an unlucky and lecherous youth by the name of Ataru Moroboshi, who is chosen by random to face Lum, the daughter of an alien invader to play a game of tag for the fate of the Earth. When he manages to win, an offer of marriage to his girlfriend Shinobu is misinterprated by Lum, and in order to maintain Earth/Invader relations he finds himself with an overzealous extra-terrestrial admirer living in the cupboard in his bedroom.
And then it starts getting really weird.
There's numerous interesting differences between the anime and manga, most notable is that Lum is portrayed sympathetically pretty much from the start. In the manga, at least in the beginning, the reader is seemingly supposed to see Ataru and Shinobu as the core relationship, with Lum being an often malicious creature. The popularity of Lum slowly changed this until the audience sympathy is aimed more at Lum, and Ataru becomes the party at fault in most cases. While the earlier stories are adapted, Lum is portrayed softer than the manga version.
Also, the first Director of the series, Mamoru Oshii has said in interviews he often did not see eye to eye with Takahashi in how he handled the series, admitting he did not care so much for the alien characters, prefering to focus on the day to day school life. This can be seen most clearly in the second film Beautiful Dreamer. Which is awesome and everyone should own. The series also provides a nuimber examples of Oshii's habit of producing stories where nothing of consequence happens (see also the individual Patlabor episodes he did).
With the shows habit of introducing more characters over time – many of which are female characters that Ataru shows a romantic interest in – Urusei Yatsura can be seen to be a precursor to the harem show.
HOWEVER, the harem show exists because shows from the 90s fail to understand what was great about shows that came before them.
Characters who are lecherous and get nowhere (or even better, get electrocuted/hit with a hyperdimensional mallet/hit by surprisingly hidden boxing glove) are funny.
Characters who are devoid of personality and have women mysteriously gravitate to them are not.
Ataru Moroboshi is a great character. Tenchi Masaki is dull as dishwater. Ryo Saeba is a classic character. Keitaro Urashima is a black hole of charisma. And don't even get me started on that piece of shit anime Rizelmine, which is like Urusei Yatsura remade for pedophiles.
Anyhoo, in terms of comedy Urusei Yatsura is the strongest Takahashi series, and I'd say because of the problems I mentioned in the Maison Ikkoku entry along with the fact a lot of talented people got their breaks here it's also probably the best anime adaptation too.