Something which I have done using microprocessors, and was part of my education, but I wonder how easy it would be to put together circuitry for traffic lights at a 4 way junction using registers and logic chips. Flip flops could hold the current state of lights (ie they are memory chips). Getting the sequencing would be tricky maybe. It might give you ideas for further things though.
A simpler idea may be to construct a binary counter using a simple digital frequency generator (one chip) to drive it, giving different digital frequency outputs. I know it is a bit 'old hat' but looking at a 7490 IC may also give you ideas.
I have not used a shift register in discrete circuitry so will have to leave that one with you.
Tracy