Lastly, the current through an LED, and therefore it's brightness, varies a lot with a small change in voltage. McWheels's circuit gets over that somewhat by regulating the supercapacitor voltage, so the current doesn't vary as the supercapacitor voltage varies. A suitable resistor in series with the LEDs could also reduce the flicker.
(McWheels's circuit has two potential weaknesses. The output voltages of the regulators might be different, meaning the highest one takes the most current. Also, the LED voltage might be too high or too low for the regulators. Slightly higher voltage regulators and a current limiting resistor for each regulator would get over both those issues. It could also be that the regulators are already running as constant current devices, in which case the exact type is very important.)
Fair points. The notes below the photo of the circuit mention the resistor I actually do use, but it's not in the diagram. My aim is to drive the LED(s) to the standard Vf @ I, and the use of LDOs is so that when the cap drops below the ideal voltage, it keeps on going rather than cutting out like a LM314.
The supercaps are also not that efficient for charge and discharge speed (ESR, I think), which actually works to my advantage, since they stop the caps being too much of a black hole at startup.
