According to flightmemory.com, I'm at 293 airports. I've only made it to that amount because I deliberately route many flights just to check off another airport (merely arriving/departing once fits the bill).
WRT the OP's question, I've visited each location of the fifty busiest airports in the world (based on passenger traffic), but have only visited DOH from XDOH (while just transiting the current DOH).