I noticed that one of the lights on my binary clock wasn't lighting up anymore, and another one was only lighting up very dimly. Thinking that maybe it would be a simple fix to replace some LEDs, I popped it open and have a look inside. It was pretty easy to open up with just 4 tabs to unclip.
Everything is all on one board with surface mount LEDs (labeled D7 - D26) instead of the regular LEDs I was expecting.
The dead diode is D24, and the fading one is D18.
The six 1N4001 diodes do some rectification of the 9VAC input power. I haven't traced out the circuit yet to see how they're arranged.
A PIC16C505 microcontroller is the heart of the binary clock. The other chip above the PIC controller (U2) is a 5V voltage regulator, which provides the power to the PIC controller.
I wonder how hard it would be to find some LEDs to replace the dead ones with.
Update: Traced out the arrangement of the diodes and this is what it seems to look like.
It's a bridge rectifier taking the 9V AC input and doing full wave rectification