I don't know what the application for your circuit is (what is it being used for) but maybe look at circuits used in solar garden lights, like these (there are 2 different circuits described on this site): http://www.talkingelectronics.com/projects/SolarLight/SolarLight.html
These circuits include a solar charging circuit for a 1.2V cell, an oscillator that boosts the voltage to drive a white LED, and a circuit to cut off the LED during daylight. If you're looking for continuous operation (day or night) the solar cell and photocell circuits could be taken out.
These solar light circuits don't have a current limiting resistor on the LED since it's driven with pulses, making it more efficient, and it's a low power circuit running from a single AA cell.
That's quite an old school piece of work! Unfortunately it won't easily fit into my head. For the second circuit:
The inductor produces a voltage (in the form of a spike) that is higher than 2.1v to illuminate the high-bright yellow LED - in fact it is much higher but the LED converts this energy to light.
You cannot measure this voltage with a multimeter as the voltage is produced in the form of spikes.
If you remove the LED, a CRO will show the spikes are higher than 40v.
Something tells me that if I replaced that battery with an alkaline AA cell, the LED will light up from brighter to dimmer as the battery depletes (1.6 -> 0.8 V). Is this the case ? What sort of dimming percentage should I be looking at for the whole battery life ?