I think what you are looking for is a WiFi router that supports a technology called WDS, or Wireless Distribution System.
Here is a bit of background on WDS
A wireless distribution system (WDS) is a system enabling the wireless interconnection of access points in an IEEE 802.11 network. It allows a wireless network to be expanded using multiple access points without the traditional requirement for a wired backbone to link them. The notable advantage of WDS over other solutions is it preserves the MAC addresses of client frames across links between access points.
- An access point can be either a main, relay, or remote base station.
- A main base station is typically connected to the (wired) Ethernet.
- A relay base station relays data between remote base stations, wireless clients, or other relay stations; to either a main, or another relay base station.
A remote base station accepts connections from wireless clients and passes them on to relay stations or to main stations. Connections between "clients" are made using MAC addresses.
All base stations in a wireless distribution system must be configured to use the same radio channel, method of encryption (none, WEP, or WPA) and the same encryption keys. They may be configured to different service set identifiers. WDS also requires every base station to be configured to forward to others in the system.
WDS may also be considered a repeater mode because it appears to bridge and accept wireless clients at the same time (unlike traditional bridging). However, with the repeater method, throughput is halved for all clients connected wirelessly.
WDS may be incompatible between different products (even occasionally from the same vendor) since the IEEE 802.11-1999 standard does not define how to construct any such implementations or how stations interact to arrange for exchanging frames of this format. The IEEE 802.11-1999 standard merely defines the 4-address frame format that makes it possible.
Here is a list from CNET of commerically available wireless routers that support WDS. I am partial to TRENDNet routers, but that is just my personal preference
CNET Reivew of Wifi Routers that support WDS
Also, if you do decide to implement WDS be sure to buy several of the same model of router. Like the above description says, WDS compatibility is spotty between devices from different vendors, and sometimes even different devices from the same company.
How many routers you will need to purchase all depends on how big each floor is. With your current setup, does a single router cover all points on that floor without issue, or is connectivity spotty in certain areas? If connectivity is spotty, I would recommend either trying to place each router somewhere close to the middle of each floor so the radius of connectivity is much higher per floor, or buying 2 or more routers per floor and placing both routers in the middle facing away from each other so each covers 180 degrees in one direction and the other covers the other 180 degrees.