Common formulas

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谢谢 xièxie thank you (absolute standard, use anywhere)
谢谢你 xièxie nǐ thank you (to you) (more personal)
谢谢您 xièxie nín thank you (polite) (for elders or pro contexts)
多谢 duō xiè many thanks (literally "much thank")

For deep gratitude

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非常感谢 fēicháng gǎnxiè thank you very much (formal, letters and speeches)
太感谢了 tài gǎnxiè le thanks so much! (warm, spoken)
麻烦你了 máfan nǐ le sorry for the trouble (and thanks) (acknowledges their effort)
辛苦了 xīnkǔ le thanks for your hard work (colleagues, teammates)

Responding to a thank you

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不客气 bú kèqi you're welcome (literally "don't be polite")
不用谢 bú yòng xiè no need to thank me (casual)
没事 méi shì it's nothing (spoken, friendly)
应该的 yīnggāi de it was the right thing (modest, common)