I'm working on optimizing the charge time. I was able to charge the 3F supercap from 2.924 V to 3.002 V in 80 minutes with a 60 W load. By a very rough linear approximation (which I know isn't accurate, but it's easy and it's not off by that much given how early this project is in its development), this would charge from 3.0 to 5.0 V in about 34 hours. This is down from the previous estimate of 100 hours.
There are still some surprising things going on. I still dont't have a proper rectifier running - I'm just using one diode and half wave rectifying. That's a lot of energy I'm throwing away. It's also a silicon diode instead of a Schottky. For some reason, the Schottky diode I bought is dropping the voltage to 0. I'll have to check that out. I have some really crazy TI "diodes" that only drop something like 26 mV, and I plan to try them soon. It would be great to make a bridge of these and to keep all the power I can.