I got my lazy ass up this morning and did some quick measurements. They more or less look what I expected assuming that the AS657G provides just some fixed PWM step-up conversion.
I guess the step-up works at 400kHz with a 20% DC or so. At 1.5V there are ~750mA flowing into the circuit and the average voltage at the output is 3.12V. At 1.2V, the current is reduced to ~500mA and the average output voltage is reduced to 3.04V. In like 80%, the voltage is above 3V though and slowly rises from 3.3V to 3.4V (1.5V supply) or from 3.2V to 3.3V (1.2V supply).
I also tried to measure the average voltage drop at the 1.5Ohm resistor and the diode with a DMM (Fluke 87V) but I'm unsure if I can trust these measurements due to the high frequent period and shape of the voltage.
Anyway, at 1.2V supply, I measured an average voltage drop of 2.684V at the diode and 0.1036V at the resistor. At 1.5V supply, I measured a drop of 2.7V at the diode and 0.14V at the resistor. Obviously neither do these values add up to what I measured with the scope (where the average values where like 0.3V higher) nor do the current values look plausible. Like 140mV/1.5Ohm would mean an average current of only 93mA which seems much too low. Again, I think the bandwidth of the 87V might not be sufficient to accurately measure the average voltage drops here, so I think the only thing I can see from this is that the input voltage is somewhat proportional to the average current. So higher battery voltage means higher output current and brighter LED.
Anyway, I guess the maximum current surely can't exceed 300mA since this would mean a 0.45V drop at the resistor leaving a maximum (!) voltage drop of ~2.95V at the LED. Actually, even assuming 80% efficiency of the step up, the average current can be >200mA for the 1.2V case as (1.2V*520mA*0.8 )/3.04V = 164mA.
While I was at it, I tried how much I could lower the voltage without the LED stopping to work. Surprisingly it worked down to 0.2V but of course the brightness dropped with the voltage, so it was very dim at this point. The good thing is that this flashlight will work down to 0.9V for sure, bad thing is that it will exhaust a rechargeable battery which might damage it.
Well, yeah, not the best circuit obviously. Which is the shame as the flashlight itself is rather nice.