Oh, I ordered two super caps today and a solar panel to play with them. The caps are rated at 10F @ 2.5V. I planned to use two in series to get 5F @ 5V. I figure these would last much longer than the five seconds I need.
My only concern now is that because I am using a Pololu switch, I need to figure out a way to make a solid state normally open switch that I can "close" to activate the Pololu switch. In other words, the circuit would typically use a push button switch which turns on the circuit when closed. But I need to do this without any physical switch, just something that determines when the capacitors are sufficiently charged and somehow "closes" without movement.
I need my board to power on about every 20 minutes, then go to sleep.