Table 1-1: ESP8266EX specifications
Operating voltage 3.0 V ~ 3.6 V
1.4.1. Power Supply
Digital Power Supply
ESP8266EX has two digital pins for power supply, pin11 and pin17. For digital power supply,
there is no need to add additional filter capacitors. The operating voltage range of digital
power supply pins is 1.8 V ~ 3.3 V
Analog power supply
ESP8266EX has five analog pins for power supply. including pin1, pin3, pin4 which provides
internal power supply for internal PA and LNA respectively, and pin28, pin29 which supply
power for internal PLL. The operating voltage for analog power supply pins is 1.8 V ~ 3.3 V.
This is why I've been half guessing the voltage range that the device is comfortable working at. To be conservative I've quoted the spec. to people on these forums as being between 3V~3.6V. But I've also made the less guarded statement that it might still operate reliably below 3V.
If, for instance, it can be run reliably at 1.8V then that would potentially double the battery life when powered by a Li-Ion cell and efficient switch-mode regulator.