That got me curious about digital clock chips.

A quick search netted me this: https://www.ablicinc.com/en/doc/datasheet/real_time_clock/S3...

They specify a typical current of 0.25 uA at 3V supply, that's 0.75 uW power. So an energy of 18 uWh over 24h.

Same ballpark, but does not move gears (just talks on the "phone" ;-)

The STM32L0x1 series has just a bit more at 0.9 uW (stop mode + RTC), but it gets you a full microcontroller.

I am unsure if we have enough of an energy budget to get us some display (e-ink likely) (and the MCU also needs some extra energy for the updates) though.

I think for an apples to apples comparison you'd need to include also some display/indicator system too, which for electrical clock would account for significant portion of the total energy use.