If you liked
Ginga Eiyuu Densetsu
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...then you might like
Taiyou no Kiba Dagram
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While LOGH is in a league of it's own, Dagram has a mature story with deep intrigue which steps into a political realm with some intersting criticism and actual real feel in which few anime series dare to go. If you're into a bit of a mature storyline and don't mind the physics innacuracies or older animation, do give the other show a try.
If you liked
What's Michael?
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...then you might like
Itou Junji no Neko Nikki: Yon & Muu
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Seinen and slice of life stories about cats. Both mangakas portrayals of the cats are rather realistic and done in the old school way even if both do go separate ways storywise.
If you liked
Giniro no Kami no Agito
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...then you might like
Hoshi wo Ou Kodomo
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Both movies try to reproduce the Ghibli feeling. Character design-wise, fantasy setting and more. If you enjoy one, you'll certainly enjoy the other.
If you liked
Ranma ½
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...then you might like
Urusei Yatsura
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Urusei Yatsura was the first big series and success of Rumiko Takahashi, it was on Urusei Yatsura where she developed her clever and wacky classic comedy stories that she would also use later on Ranma. Both series went to being highly influential on the anime industry and comedy anime was never the same after them, especially Urusei Yatsura. Needless to say that if you enjoy one, you'll also enjoy the other.
If you liked
Urusei Yatsura
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...then you might like
Maison Ikkoku
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If you want to find more Rumiko Takahashi humour and want to see a story with beginning, middle and ending with a more serious romantic approach then search no more. Maison Ikkoku is unmatched on the Romantic/comedy/drama anime field.
If you liked
Versailles no Bara
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...then you might like
Ginga Eiyuu Densetsu
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Legend Of Galactic Heroes may not be a shoujo like Rose Of Versailles so the romantic aspect is missing, but pretty much everything else about the series are similar, I may risk and say that LOGH went to take some aspects and influences of Rose, mainly the political and social critic of the aristocracy. The dramatic aspect and overall feeling throughout the series are pretty similar too not to speak that even the narrator has the same feel.
If you liked
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...then you might like
Roujin Z
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In terms of entertainment and overall feeling, I can relate this one with Otomos Stink Bomb. The little bits of well thought comedy, allied with political critic are very similar.
If you liked
Cosmo Police Justy
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...then you might like
Terra e...
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Justy looks a lot like Blue. Both are very strong espers and the character designs, hairstyles and overall art are awfully similar. It's like Justy was a reincarnation of Blue in an alternative universe. Even Cosmo Police Justy animation seems to be influenced by the "Terra e..." movie.