You are right, ESP8266 doesn't actually have EEPROM - the EEPROM library is just faking it apparently. I just used some code from a previous project of mine - on Arduino Nano - which actually doesn't work on ESP8266. Need to update that part. (it does still increment the counter while it's plugged in, though)

Look into EEPROMAnything, and store the number in a long.

I've used EEPROMAnything on STM32duino and it worked well - I think it would work for ESP8266.