Unfortunately, my original plans with a simple joule thief just didn't want to work, so I redesigned the whole thing. I focused on a higher output voltage, allowing for a lower capacitance and less strain on the joule thief. To accomplish that, I had to make a high voltage joule thief - a joule thief with a third, high voltage coil.

I wound the transformer and, sure enough, it worked like a charm, so the only thing that was left was to shoot the circuit in action and document the process of making it. 

The circuit now produces an output voltage of 22 Volts with the capacity of 10 000 uF, which makes for 220 mJ of energy, which isn't only enough to burn LEDs, but is also enough to weakly spot-weld wires to each other, as I have accidentaly found out.

You may notice a few updates: an updated components list and building instructions. There should also be a link to my youtube demo of the circuit.

Good luck to the other contestants for the Coin Cell Challenge!