Some influential anime that I have watched over the years:
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Dragon Ball Z. My all-time favorite anime, mainly because I grew up with it and it was the anime that introduced me to anime. Been a fan of the series since 1999 and am still a huge fan today. Very influential anime that set the bar for all shonen anime after it.
I also love
Dragon Ball and, to an extent,
Dragon Ball GT.
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Astro Boy (2003). The anime that made me become a fan of Astro Boy! I first watched it sporadically back in 2004 and later revisited it in February 2011. Though I consider it to be my least favorite overall of the three Astro Boy anime, I think the final 6 episodes is one of the best stories I've seen told in an anime. I also love the '80s series and still have yet to see the '60s series in full. Great series and a great fanbase!
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Hunter X Hunter. My favorite on-going anime at the moment. Great storytelling and great characters (and character development). I prefer
the 1999 anime a tad bit more (the opening is in my top 5 favorite anime openings), but the 2011 anime is very good as well.
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Now and Then, Here and There. Wow, where to begin. Just typing the name of that anime brings back a lot of memories. I first watched this anime back in September 2008 on the Sci-Fi Channel (now the Syfy Channel). It holds a special place in my heart and is what I consider to be the greatest anime I have ever watched.
The anime deals with putting children into war and the consequences behind that. It's a very emotional and dark anime. If you are 16 years old or over, I highly recommend watching this series. All 13 episodes are available free online if you want to watch it.
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Nobody's Boy: Remi. Based on
the 1878 French novel by Hector Malot. Don't let the opening fool you. This anime, like Now and Then, Here and There, is an emotional roller coaster that some consider to be one of the most depressing anime ever. If you can get past the drama that this series holds though, you'll find this unknown gem to be a very enjoyable and satisfying experience. Sadly, the series is no longer available in full anywhere legally to watch online. When I watched this series, it was up for a limited time to stream on Anime News Network back in 2010. It was taken down recently though, most likely due to low view counts, which is quite unfortunate.
I could post more anime, but I'll leave off there for now.