I added better readings above.
I am now trying to use 4xAA (at about 5.4V initially) through an AMS1117 on the esp-12. I measured almost 3mA from the battery to the 1117 when there was no load (the LED was removed from the 1117 board, it was more like 7mA with it through a 1K resistor).
If this is a reflection of what one can expect from using a regulator then it is not promising, I hope a proper LDO does a significantly better job to make it an overall win.
Here is an example of a run with 3xAA (2300mAh NiMH) batteries without any regulation:
The voltage started at about 4.1V* and dropped below 4v after two days. After it hit 3.7v it started a nosedive and would have been depleted in less than a day.
If we supply regulated 3.3v then on the esp we save between 650uA and 80uA. How much will the LDO dissipate? How will this change if we use 4xAA(5.4-4.8v) rather than 3xAA(4.1-3.6v)?
I am ready to test this.
(*) Vdd is measured with a 12 bit ADC so the often max 4.094v read is an indication of an overflow. But I do not expect it gets much higher - I see just under 4.3v from 3xAA fresh out the charger with no load.