
lot of times somebody helped to me on this forum, so I decide share my tests and tries with battery powered sensors.
Datasheet and everybody says abou maximal voltage on ESP8266 is 3.8V but that is not absolutly true.
I made lot of measurements and experiments with 18650 Li-ion batteries as power source for ESP8266.
As protection I added shotky diode between ESP and battery (shotky pull-down voltage of battery) but consuption of diode was higher than consuption of ESP when you connect in on higher voltage as is recomended.
When I connected current meter between power source and ESP, I saw some dependance between voltage and current consuption. On higher voltage (4.5V) is higher current consuption. I tried compare discharge li-ion battery from 4,2V to 2,5V with and without compensation diode.
Result is.. do not use diode

If you use very good LDO regulator, with veeeery low internal consuption.. That is better solution.
I'm using sleep mode in ESPEasy and whole concept of ESPEasy it's very good for me, so I ended my own programming of project and using ESPEasy on battery sensors.
Maybe somebody will help this.
Here you are my small results with different batteries and measurements:
(on ESP-01/02 I've made hack - soldered GPIO16 pin to Reset for use sleep mode)
sorry for grammar mistakes in my English