How to run ESP8266 for months with Coin cell?

I am sorry to cut you off your idea, but using coin cell to power an ESP is not a good idea. I will put forth my reasons below you can decide later

1. Coin cell is meant for low current applications it can source a maximum of 10-20mA based on the type. Whereas an ESP8266 is a power hungry module it requires as high as 80-100mA when it is trying to connect Wi-Fi. So no coin cell can source that current. 

2. Coin cells have a operating range from 2V to 3.3V, so to use the complete juice out of the cell we have to drain it till it reaches 2V. But ESP8266 has an operating voltage  above 3V. Hence a DC-DC (Step-up) converter becomes mandatrory 

3. Using a DC-DC converter to utilize the battery and provide more current through capacitors will be highly in-efficient and would drain the battery faster than it is supposed to.

Considering all these I have never used a coin cell to power an ESP8266, instead I always use 3.7V 200mA lithium Polymer cell since they have the same form factor and avoids extra circuitry. These batteries are commonly used in mini drones  

While it is true that jayant has a point, you can still power your ESP8266 with a coin cell  and there are people who have done it.

The problem of high current requirement can be tackled by using a super capacitor.

The ESP8266 needs about 80mA only for a fraction of second when it tries to connect to internet this can be sourced with help of a super capacitor in parallel with a coin cell. This way the capacitor will get charged slowly from the coin cell and when there is demand by the ESP8266 it discharges and thus sourcing enough current for the ESP. 

Remember to have a resistor in between the cap. and battery to limit the current. Also there is no way we can void a DC-DC converter so it stays mandatory 

Let me know how this works for you

Hi, you can use 18650 cell with a 3.3 voltage regualtor or you can use a small 3.7v lipo battery.