Doing a programming language/compiler was very educational and did a lot to demystify computers for me. I would also second the recommendations to write a game

Doing a programming language/compiler was very educational and did a lot to demystify computers for me. I would also second the recommendations to write a game. If for no other reason than the fact that it is some of the most fun programming there is, you usually get to flex muscles you'd never get to in a boring web dev job.