"How many days does it have been (since we started dating)?"
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The suffix ~째 = ~th
첫째 딸 = first (eldest) daughter
뚤째 아들 = second son
셋째 자식 = third child
I don't think there is a perfect match in English for 며칠째, but I think you know by now where it leads:
며칠째 = how many 'th' of day