How about this?
8 74xx30's (8 input NAND) with each output feeding one of the inputs
of the other stages. The remaining input of each gate is connected
to a pushbutton whose other side is grounded and the gate side pulled
up to Vcc through an appropriate resistor (say, 2.2K). The circuit
is self-debouncing and the last button released governs the output.
Basically an 8-stable flip-flap-floop...-flop. Initial state is
theoretically random, but probably could be forced with a small
capacitor across one of the buttons.
If you expand much beyond this, note that fanout issues may rear
their ugly heads.
Cheers,
Chuck