This "Joule Thief" circuit, using a battery that won't work in any other device, can power a blue or white LED for approximately 8 days straight before it stops working. It could probably power a red LED for at least 10-12 days. You could also use it to power a microcontroller (such as an AVR or PIC MCU).
This is an entry in the LED Contest with Elemental LED and the Hurricane Lasers contest, so if you like it, vote!
Step 1: Why Is It Called a Joule Thief?
Basically, what the Joule Thief does is it "steals" every last Joule of energy from your batteries. The explanation of what a joule is can be found on the Wikipedia page here. The quickest explanation I can think of is that a Joule is a form of energy that, in the way we use it, can be used to power electrical devices. I won't go into any more detail here because it gets very complicated.
Step 2: Parts List
1x 470uH inductor (looks like a fat resistor)
1x 2.2K resistor
2x 1K resistor
2x basic NPN transistor (2N3904, 2N2222, 2N4401, etc.)
1x 1000pf (same thing as 1nf or 0.001uf) ceramic capacitor
1x AA battery + holder (or any other 1.5 volt battery)
As many LEDs as your heart desires.
Step 3: Make It on a Breadboard!
Step 4: Make It Permanent!
Step 5: It's Done!
You can also use a 2x AA battery pack instead for enhanced brightness. I'm not sure how many batteries you can go up to until the LEDs burn out, but you can sure test it yourself! :)
Step 6: BONUS! Use It to Power Your Projects!
You could also probably make a super-high-power Joule Thief power supply with a 9 or 12 volt battery instead!