A Joule Thief is also unrelated to running an LED continuously as 20mA.Well, here then, is another Big Clive video, describing a circuit called the "Joule Thief", which runs an LED (forward voltage ~2.5v) on a nearly completely depleted AA cell (0.5v measured):
It works by using flyback voltage from a tiny transformer hand wound on a ferrite bead, and uses only 1 common NPN transistor and a 1K resistor...surely you can make a single LED run for a long time on a rechargeable battery with this circuit. But you can't defeat the laws of nature; the total energy capacity of the battery must be enough to drive the LED at your desired current for the period you want ...this circuit just squeezes the last bit of energy from a battery, and makes it useful, even if the voltage is too low to be used directly.
Hope you find this useful. For more info, a Google search for 'joule thief' might be instructive...
Please stop posting answers until you are sure that they actually relate to the question.