If you liked
Tengen Toppa Gurren Lagann
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...then you might like
Planet With
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Disregarding Trigger works that went into another direction, Planet With is as close to TTGL as I have ever seen. Both give a cosmic meaning to the phrase "surpass oneself", intertwining action packed mecha battles with colourful characters arcs. They diverge though due to their relative positions in the anime medium. TTGL the longer forefather series focuses on raw energy and will, both in themes and in animation thanks to Gainax, while Planet With is a reflexion on TTGL: shorter, the plot and animation are more streamlined, thanks to JC Staff, while also exploring the themes of love and forgiveness.
If you liked
Hikyou Tanken Fam & Ihrlie
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...then you might like
Kaze no Tairiku
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Both anime have a small team of specialists struggling to survive the dire consequences of ruin exploration. While different in tone -- Fam & Ihrlie light and playful high fantasy, Wind Continent more serious and dark like Conan -- both anime should satisfy the traveler in search of new adventures.
If you liked
FLCL
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...then you might like
Rolling☆Girls
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In both form and content this is the closest thing I've ever seen to FLCL since watching it 10 years ago. This anime is of course lighter and more girly (and longer), but it captures the bursting energy of youth with catchy rock songs, gorgeous art and animation, and an original alternate-reality setting. One difference is Rolling Girls focuses more on travel, but as a trade-off it also has way more characters than FLCL.
If you liked
NieA Under 7
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...then you might like
Shoujo Shuumatsu Ryokou
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Both are post-apo atmospheric anime that go full SOL and leave the story in the background. The notable difference is SSR is more relaxing and cute with high production values, while NieA is more a gag oriented budget anime.
If you liked
Slayers
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...then you might like
Hikyou Tanken Fam & Ihrlie
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Same 90s fantasy vibe (art and chara design) both are light-hearted with a more serious main plot involving evil wizards both procure a viewing experience that isn't like anything we see today
If you liked
Mimi wo Sumaseba
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...then you might like
Kotonoha no Niwa
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They're both realistic love stories that involve highschoolers and craftmanship with nice art. Of course Ghibli's art is different and older than Shinkai's, but the atmosphere is kind of alike.
If you liked
Natsume Yuujinchou
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...then you might like
Pupipo!
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Same premise of a young person getting into trouble because they can see spirits. Both heroes receive help from a powerful good spirit that becomes their familiar. The two shows manage to convey a lot of emotions. As for the differences, Pupipo is a tad more comedic and childish, it's also shorter.
If you liked
Sora wo Miageru Shoujo no Hitomi ni Utsuru Sekai
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...then you might like
Kyoukai no Kanata
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Both shows are action shows made by KyoAni, which gives them a similar atmosphere. A lot of the visual effects used in Kyoukai no Kanata were developed in Munto. The plot revolves around a human girl with powers and her encounter with a demonic boy.
If you liked
Dead Leaves
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...then you might like
Kill la Kill
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Kill la Kill is basically Dead Leaves in school. Having some staff members in common, character design, animation, direction give the same feel. Both works are also fast-paced and filled with over-the-top action. They both depict fascism and more generally the ugly side of life (dirtyness, physical violence, verbal abuse) in a non-politically correct way.
If you liked
Dead Leaves
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...then you might like
Trava: Fist Planet
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Dead Leaves and Trava have a lot in common: a dystopian sci-fi premise, an artwork and an animation style that give a frenzied and organic feeling, the focus on few unconventional characters, the comedic tone, even the duration. Only big differences are the plots and the fact that Trava was made two years before Dead Leaves, making it a ever so slightly less fluid.
If you liked
Tonari no Yamada-kun
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...then you might like
Lucky☆Star
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One thing I like about Lucky Star is how the short comedic segments recreate something of the repetition and randomness of everyday life, but in a refreshing kind of way. Well this format was basically invented by Tonari no Yamada-kun. Tonari no Yamada-kun doesn't focus on cute girls and school life, but on the whole family and is more deep. Overall Tonari no Yamada-kun's art is simplier than LS, but it has moments of complex and realistic art/animation that can't be found in LS.
If you liked
Shinseiki Evangelion
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...then you might like
Top wo Nerae! Gunbuster
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The first direction of Anno, Top wo nerae shares a lot of aspects with NGE (themes, characters, direction...), while in a denser package. A must-see for any SF anime fan (and a "watch-it-right-nao" for Anno fans).
If you liked
Shinseiki Evangelion
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...then you might like
Fushigi no Umi no Nadia
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The same team worked with Hideaki Anno, so NGE and Nadia really have the same feel to it. Plus the bottom line is pretty much the same, a mix of fathomless lore, technology and how they interact with the human condition.