Router setup for network with more than 255 devices?

Howdy! We're setting up a network for an Ethernet-enabled art installation and are running into issues. We've got 300+ devices connected to our router, and each of them asks for an IP via DHCP. However, our router will only let us assign IP's in the 192.168.1.1 to 192.168.1.255 range (and not, say, 192.168.1.1 to 192.168.5.255), and so we're running out of IP's.

We're artists, not network engineers, but we know that it's possible to have a network with more than 255 devices. Duh! How might we go about this? Thank you!