With two sets of stacked super caps, the total storage is brought to 60F.
The below is the current draw of each major component. This doesn't encapsulate everything, especially with changing supply voltages, but it is a good show of how the LEDs have the majority of the current draw.
555 Timer: 0.25mA
4-bit binary counter: 1uA
4-16 bit decoder: 8uA x 2
Single LED: 27.5mA
I selected the regulator to be 2.75V (just under the 2.8V rating of the caps). Since the forward voltage of the LEDs is 2V, this device has a useful range of only 0.75 volts. Once below 2V, the LEDs will no longer light up and the device will slowly bleed out the remaining energy.
Charging Time and Run Time
Upon powering the device with the charging coil, the caps quickly charge to 90-95% of the max voltage. I originally thought this was "close enough" to being charged. When removing from the supply after less than a minute of charging, the device would run out of power within a very short period of time.
Charging it for longer and monitoring it shows that the voltage quickly ramps up, but flattens out near the peak and takes quite a while to actually reach 2.75V. When giving it a long time to charge completely, the device was able to run for about four hours.
Possible ways to increase run time:
-increase supply voltage (limited by capacitor specs $$$)
-more capacitance ($$$)
-choose LEDs with lower forward voltage
-run LEDs dimmer (larger series resistance)
-find a timer with lower current draw (if available)