What is the maximum number of clients that can connect to AirPort Extreme?

50 Wireless and 150 Ethernet, although it is possible to configure the AirPort for even more Ethernet clients if needed. Only wireless devices can connect to the Guest network, if it has been enabled.

Note the the wireless is going to slow noticeably at about 20 clients and slow even more as additional clients connect. So although 50 wireless devices can connect, things are going to be mighty slow if all of them are active at the same time.

The current model supports up to 50 users. http://www.apple.com/airport-extreme/specs/