You shouldn't automatically assume you need one AP per floor --- wifi signals pass through floors too, so too many APs will just interfere with each other. For comparison's sake, I have a 3-story house and I have good signal throughout the house (and OK signal on the outside porches) with two APs, both on the second floor and on opposite sides of the house. My place is smaller than yours (~1000 sqft/floor), but on the other hand its 1910s brick, wood lath and plaster construction, so way less wifi-friendly than most newer homes.
I'd try starting with 2 APs, placed pretty far apart, and check your actual performance before deciding you need more. And keep in mind that high performance is not about how many APs you have, it's more about the quality of the backhaul setup. The manufacturers will try to tell you that "wireless mesh" solves everything, but the only thing it solves is not having to run wires; you pay for that convenience in lost performance.