Here's a little slideshow on how to build a simple, small, Joule Thief on a scrap of PCB.
The toroid is a tiny one scavenged from a CFL base power supply, stripped and rewound with fine magnet wire. Use 20+20, or 16+16, or experiment with winding numbers. Same for the Base resistor value.
This unit will drive 3 or maybe even more White LEDs in series, from a depleted button cell or AAA battery. It works best with input voltages under 1 volt, and can also drive small pulse motors, etc.
Music is "The Messenger" by Silent Partner, from the YT free audio library. Thanks guys and gals!