The NE555-timer is quite a cool little thing.

More common operational amplifiers are also worth looking into (TL072 is one of my favorites).

Lastly, the 7400-series logic chips are SO interesting. If you need the programmer's version of a jigsaw puzzle, try to build somewhat advanced designs using those. It's very fun and a good learning experience (although hard!).

This guy is building a computer on a breadboard – very intersting indeed: https://www.youtube.com/channel/UCS0N5baNlQWJCUrhCEo8WlA