No microcontrollers needed here! I got carried away playing around with some 7490 decade counters and ended up building a digital clock with stuff I have around the lab.
The first trick was getting an accurate 1 second time base. There are a few options here. You can divide the 60 Hz AC line frequency (in the US) down to 1 Hz or you can build a crystal oscillator and divide that frequency down with decade counters.
For this clock I ended up taking a 1 Hz oscillator circuit out of an analog novelty Mickey Mouse clock I had laying around. There are a few different ways you can wire these oscillator circuits up explained very nicely here. It took some experimentation to get it working.
With my clock I didn't have to run the outputs through diodes or transistors. Each output is ½ Hz and simply wiring them together worked just fine. I powered this oscillator off the 5 volt supply by wiring a current limiting diode (specifically an LED) across its power input. This had an unexpected "feature" where the LED blinks off every second.
The time is kept by six 7490 decade counters. The 1 Hz clock goes into a 7490 wired up as a mod 10 counter (for seconds 0-9). The output of the mod 10 counter then goes into a 7490 wired as a mod 6 counter (for seconds 0-5). That circuit is then duplicated for the minutes (0-59). The hours are then counted by two 7490s wired up as a mod 24 counter (0-23).
The tens of seconds are displayed by 3 LEDs in binary. The 7 segment displays are driven by four 4511 BCD-to-7 segment decoders. I needed thirty-three 100 Ohm resistors in total for all the displays and LEDs.