Driving 4 white LEDs independently
You could do a sort of modified high-side PWM controlled joule thief for each of the LEDs, with each having an independent PWM controlling the power though the LED. 

Simple version might look something like this...







simulate this circuit &ndash; Schematic created using CircuitLab

This is different from a normal joule thief in that...


The chopping is done via PWM rather than feedback. This lets you control the power though the LED. 
The LED is connected to the high side rather than ground. This means you will need more kick in the inductor to get the LED up to minimum voltage, but has the advantage that there is no leakage current though the LED when the PWM is off. 


Note that there is not a straight relationship between PWM and brightness here. When PWM is off, LED is off and no current flows. As you increase PWM duty cycle, you will hit a minimum current needed to light the LED. Then as you increase more, you will get brighter and brighter LED until you hit a maximum. Past the maximum the LED will start getting dimmer and you will start killing the battery, so stop increasing the PWM duty cycle when you get to the maximum brightness.

This circuit should have very low quiescent current- basically whatever the leakage current of the transistor is.