Anything from three to thirty. How much you see, how long you see it, and at what detail determine how many days you can spend seeing the sights in DC.
You can, for example, walk around the Capital Building, the White House, and the Mall in one day. But if you were to sign up for and take the various guided tours with each one, you could spend several days with each one. The museums and such along the Mall alone would take a week to do in detail.
I went to school nearby DC; so DC was a regular place for recreation for me. And there were places to go for days without repeating myself.
A better tactic for you or anyone going to DC is list what you want to see and then figure out how many days to set aside to see that list.