Analog?
- Try a 723 and build a good power supply. Yes these are old but at least as mythic as the 555, and you'll learn a lot from those implementations.
- Experiment with common cheap operational amplifiers such as the 741.
- LM3914 and LM3915 and build led bar-graph meter. as sound level indicator or battery level indicator.
- XR2206 as function generator, but I think these are not manufactured anymore.
Digital?
- Choose an 8 pin atmel µC and build a blinking light on a breadboard. Sounds silly and too simple, but you'll need a power supply and have to be able to program that thing. Make it blink at the rate of something you measure (temperature, light level,...)
- Build a rs485 interface and have 2 computers/controllers talk over that serial interface.
- 4066 and switch between audio signals.
- TLC5940 16-Channel LED Driver.
Alternatively, search for the electronics magazines of the late 1990's or 2000's. There are plenty PDF's online. A I'm sure you'll find some interesting projects that you can dissect on component level. Circuit Cellar; Elektor; Nuts and Volts mag; everyday practical electronics; ...