To prove the concept of century-long-lasting device, the proposed purely-solar-powered clock with no electrochemical primary or secondary battery should be built.
As commercial energy harvesting ICs probably won't support complex switching between various parts of the application with different power consumption, the following simplified solution is proposed:
Here, the low-consumption part has its own storage (capacitor), which is "switched" (decoupled) by a simple diode. Of course, the diode has to be carefully chosen for reverse leakage - 10s of nA may be a showstopper - a 1N4148 leaks just around 10nA at 3V reverse voltage, but we may try to find some better diode. The external leakages must be reduced, too; here an absolutely clean PCB is a must.
If extra fun is desired, the mechanical storage/primary source can be attempted.