The left button is used to put the ESP8266 into flash mode for updating the firmware, the right button is used to reset the device, effectively pulling it out of deep sleep to refresh the content on the screen, and the center button is wired to a GPIO pin that isn't used. You would only be able to use the single button for input. I don't see any other GPIO pins exposed, so it'd be tough to add extra buttons.
The ESP8266 has over a dozen GPIO pins, but any hobby kit like the ESP01, ESP12E, or ESP32 might not expose all of the pins. I can't tell which ESP module this board has on it, but if they're exposed you could solder an extra few buttons. They'll be really small solder points, though.
The ESP8266 has over a dozen GPIO pins, but any hobby kit like the ESP01, ESP12E, or ESP32 might not expose all of the pins. I can't tell which ESP module this board has on it, but if they're exposed you could solder an extra few buttons. They'll be really small solder points, though.