No specific order:
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Naruto learns about Jiraiya's death
The anime's good times. All involved characters had lots of build up and the story was written very well. That's what makes this part so emotional to me. Back then, the writers could brilliantly deliver the emotions into our hearts.
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Hinata faces Pain to rescue Naruto
One of the best anime moments ever. It shows how powerful love truly is. It means that you risk everything in order to at least try to save the person you have deep feelings for. She knew she could't defeat Pain, she knew she would most likely die, but all that didn't matter to her, nor scare her at all.
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The Ame-orphans
My favorite characters in the Naruto series. Their entire story is tragic and sad, but very well written. The characters had depth, authenticity and great personalities while their downfall is pretty much relatable in real-life, since it's not uncommon for great, kind people to turn into shattered beings because of pain, sorrow and hatred.
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Ashitaka from Mononoke Hime
I really like his entire character. His story - well, basically the entire movie, is a metaphor towards our very daily lives and we all can learn a lot from it. Throughout his journey, he learns to live with the hatred that had infected his body. He took all the hatred around him, focused it on himself. It was hard to accomplish and didn't always work out, but he gave the best he could. In addition to that, he converts the hatred into romantic feelings for San. And that is very relatable in real-life as well.
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Kazuto and Asuna "dying" in each other's arms
Maybe you call it "cheesy", but I love this romantic, yet very sad scene. Both of them were trapped in a digital world they didn't know. They had to get used to their new lives and eventually grew together. In fact, their bond became so strong that they fell in love, even though they never saw each other in real life. And at the end, when the digital prison started to collapse, they were ready to let go and die together. They were happy and thankful for the time they got to spend with each other. That's what I value a lot.
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Hotaru and Gin touching for the one and only time
Another love story. Hotaru meets Gin as a very young child the first time. They become friends, spend time with each other every year while Hotaru grows into a young woman and Gin doesn't age at all. The problem is that Gin cannot be touched by a human, otherwise he would be erased from that world and basically die. Eventually, their deep friendship develops even further into a blossom of love. But very short after that, Gin accidently touches a human and realizes that his time has come to an end. Knowing this, he and Hotaru finally can enfold each other in their arms as Gin slowly fades away.
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Zuko:
"Uncle, I know you must have mixed feelings about seeing me, but I want you to know .. I am so, so sorry, Uncle. I am so sorry and ashamed of what I did. I don't know how I can ever make it up to you, but I.."
[Iroh pulls him into a hug]
Zuko:
"How can you forgive me so easily? I thought you would be furious with me."
Iroh:
"I was never angry with you. I was sad because I was afraid you'd lost your way."
Zuko:
"I did lose my way."
Iroh:
"But you found it again. And you did it by yourself. And I am so happy you found your way here."
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Korrasami
A beautiful way to finish the tv-series. Great characters, great development. There isn't really much to say, it's just so well done. Not only was it good for the series, but also for the people who watch it and are able to relate and also get happiness, confidence this way.