Nate's toys leave me with a lot of “dead” AA cell batteries so I decided to make a Joule Thief (also here Evil Mad Scientist Labs and Wikipedia: Joule Thief), an unregulated switching power supply. It's an interesting circuit mostly because it can be assembled with so few parts.
I used a generic 2N3904 transistor. After three failures to get a working oscillator with hand wound coils, I found a tiny commercially assembled ferrite coil in a bag of parts. It has seven turns, which seems to be enough. Adding a 10uF capacitor in parallel with the resistor resulted in a brighter light.
After getting the circuit working, I read about experimenters using the back EMF from the collapsing field. Sure enough, it works. I'd like to say I skipped suggestions of protecting the LED and the collector on the transistor with a diode because I had a $1 limit on parts but really I couldn't get a reliable oscillator with a diode and 0.1 uF cap in place.
Here is my effort, mounted and driving a single green LED off the back
EMF from a “dead” battery: