How to power a white LED from batteries
Try this little circuit for a start. It runs from near-empty AAA batteries.







simulate this circuit &ndash; Schematic created using CircuitLab

Well, how does it work?


In the beginning, both the transistor and the LED are shut off.
A small current flows through the lower leg of the coil into the base of the transistor.
The transistor turns on the C-E path. Now a much higher current flows through the upper leg of the coil and the collector of the transistor.
Because the upper and lower leg of the coil are magnetically coupled, the current cannot flow in opposite directions in both legs. The small current into the transistor base runs dry.
The transistor shuts off again.
The current running through the coil cannot be shut off in an instant! That's why the coil increases the voltage at its terminals to sustain the current. That voltage is added on top of the source voltage.
Eventually, the voltage is high enough to bring the LED into conduction (and lighting it up in the process.)
The voltage of the coil decreases again.
The LED shuts off again and the process restarts.


Congrats. You have built and understood the simplest boost converter of all.