I sometimes hear people say I appreciate you coming. What's the grammar behind this sentence. Shouldn't it be I appreciate you're coming or even I appreciate your coming.

best regards

Hi Ivanhr, and welcome to English Forums.

Yes, you will hear "I appreciate you coming" very often.

Technically, yes, it should be "your coming" because (as Anonymous has pointed out) it's the act of coming that is appreciated, and it's their action.

You can say "I appreciate that you are coming" - but that relates to a future event, while "I appreciate your coming" could be present or past.

However, the "you" version is so common that it's best to not flinch - just continue to say "your" (or his, or my, or other possessives) for when it's the action that is appreciated.
Ivanhr: Many native speakers consider "I appreciate YOUR coming" as choice English; "I appreciate YOU coming" is also "correct" in regular writing and especially conversation. Some people prefer "your" because I appreciate the "coming" -- I do not appreciate "you." On the other hand, you would probably say, "I saw you coming down the street" because I saw you. I did not see the "coming." (It would be helpful not to say "you're coming" because that means "I appreciate you are coming," which is not considered correct English. )
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 BarbaraPA's reply was promoted to an answer.
Thank you both for clarifying that