I would caution users to not use 2.3 volt super caps with only 2 in series for 5 volts.
I am a firm believer in capacitors should have a 10 percent minimum margin of safety on voltage and 20 percent is much better, therefore I would use 2 of the 2.7 volt supercaps, or 3 of the 2.3 volt units.
I know the diode can drop .3 or .7 volts depending on the type, but there is no guarantee that the end users here will know which diode does what.
Also, a note for Greg, the LM7805 does have a current limit/clamp of 1 amp so the super cap charging will be regulated as far as a surge is concerned. I do not know the response time of the current limit of the LM7805.
Also, the resistor seems large to me, 470 ohms at 10 ma gives a 4.7 volt drop, and the bright white leds would be on the dim side with this value.
I am not picking on Dave, just trying to make sure if someone uses his circuit that they understand some variables that were not mentioned here and going to Dave's site would be better than trying to make the above circuit meet their expectations if the correct parts were not used. The site does need a note on the proper diode selection as Dave assumes .7 volts for the diode voltage drop and no mention of a particular type/part number to guarantee this.