The number of devices that can be paired is practically unlimited – as long as you have enough storage for every device's MAC address and link key.
In a piconet, one master can actively communicate with seven other devices (limited by a 3-bit address), and up to 255 devices can be part of the piconet but inactive ("parked").
Bluetooth communications are not limited to building piconets, however. More often the connection is between exactly two devices (for example, file transfers between two phones). In this case, since the full 48-bit MAC address is used, the master could juggle many such connections at once, within hardware limitations.
Another possibility is a scatternet, where multiple piconets are linked by a shared device. Wikipedia says, however, that "currently there are very few actual implementations of scatternets due to limitations of Bluetooth and the MAC address protocol."
If you choose your own hardware, also consider ZigBee, which is another wireless networking specification with less restrictive limits and more efficient power use:
Because ZigBee nodes can go from sleep to active mode in 30 msec or less, the latency can be low and devices can be responsive, particularly compared to Bluetooth wake-up delays, which are typically around three seconds. Because ZigBee nodes can sleep most of the time, average power consumption can be low, resulting in long battery life.