Tricky bit is getting the PIC to start from 1.2-1.5v in the first place.
I wasn't sure that this would work since it looks suspiciously like a perpetual motion machine. I think it has to transfer energy from the battery into the series capacitor if it is to be sustainable.
I did wonder about the idea of making the on-off switch a dpdt with a low leakage capacitor across the battery when the unit is switched off, and in series when switched on. And then a boost converter to keep it topped up during use. The PIC will start at >2v and keep running once started a little lower than that.
There are clock outputs at 32kHZ(unbuffered), and buffered 8192Hz and ~50Hz available
I think I like the Joergs idea of the 74AUP1G14 low power Schmidt trigger inverter best. Cheap and probably cheerful. There is something appealing about getting low current stuff to run off a single 1.5v cell efficiently.
The challenge here is to do it cheap, compact, with easily available bits and a low parts count in a form that will work reliably for a home constructor. There are quite a few hobby PIC circuits that would benefit from a single 1.5v battery.
Regards, Martin Brown