You can install the latest version (10.6.8) of OS X Snow Leopard on your Mac mini.
This system can run the last version of Mac OS X 10.6 "Snow Leopard". However, it cannot boot in 64-bit mode nor does it support "OpenCL" or "Grand Central Dispatch". It also cannot run versions of Mac OS X more recent than 10.6.8.
You can buy OS X Snow Leopard here: http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard