Yes, it's the infamous Joule Thief, in Instructable form! For those of you who don't know, the Joule Thief is a tiny little circuit that allows you to drive a white or blue LED from voltages as low as 0.5 volts. You think those batteries are dead? Don't throw them out yet! Hook them up to the Joule Thief to squeeze every last drop of energy out of them!

The idea and circuit came from this Make weekend project. Why don't you pay them a visit?

Step 1: Parts and Tools

For this project you will need very few tools and parts, as you will see in the picture below.
But for those of you who like it in text, here it is:

Helping Hands (Optional)
Soldering Iron
Solder
A Blue or White LED (Other colors are fine, too)
2N3904 Transistor or equivalent
1k Resistor (Brown-Black-Red)
Toroid Bead
Thin wire, two colors (magnet wire works)

You can get the toroid and transistor from a dead CFL; the transistor is usually labeled 13002.

Also, if you use a 2N4401 or BC337 transistor, your LED will be brighter because they can handle more amps.

Step 2: Wind the Toroid

The first step is to wind the toroid. I found mine in an old computer power supply, and it works fine for me. Toroids are donut-shaped objects like in the picture, and can be attracted by a magnet.

You can find toroids in a few places. Old computer motherboards, XBOX and X360 motherboards have them (don't take them unless it's dead!). You can find toroids in computer power supplies, or you could buy them at your nearest RadioShack.

Take your two strands of wire, and twist the ends together. You don't have to do this, but it makes winding a little easier.

Thread the twisted end through the toroid, then take the other two ends (Not twisted together) and wind it once around the toroid. Don't twist the wires; make sure that two wires of the same color are not right next to each other.

Keep winding, making sure you wind the coils tight. It will still work if they are kind of loose, but it is better to have them tight.

Ideally, you want about 8-11 turns on your toroid. Even if you can fit more, don't put more on. Make sure the turns are spaced evenly around the toroid.

Once you wind around the whole toroid, cut off the extra wire, making sure you leave a couple of inches for soldering.

Strip some insulation off the wires, then take a wire from each side, making sure they are of the OPPOSITE COLOR. Twist them together, and then you're done with the toroid.

Step 3: Solder It All Together

Now we are going to solder the whole thing together. You could put it on a board if you like, but in this Instructable, we're going to free-hand solder it. Or whatever you call it. You can follow these written instructions, or take a look at the pictures. They explain it very well.

First, take the two outer leads of the transistor and bend them outwards a little ways, and bend the middle lead backwards. Bend the LED's leads outwards, too. You don't have to, but it makes it easier to solder.

Take one of the wires coming from the toroid that is alone; that is, not twisted together with another wire. Solder it to one side of the resistor. Solder the other end of the resistor to the middle lead of the transistor.

Take the other single wire from the toroid, and solder it to the collector of the transistor. Solder the positive side of the LED to the collector as well, and solder the negative side to the emitter.

All that's left to do now is solder an extension wire to the negative side of the LED. Take a piece of that wire you had earlier and solder it to the transistor's emitter.

Step 4: Try It Out!

And that's it! You're finished with your Joule Thief. Touch the twisted wires coming from your toroid to the positive side of a battery, and the extension wire to the negative side. If all is working well, the LED will light up! If not, try using thinner wire on your toroid. I've included a schematic in the pictures if you prefer.