http://spritesmods.com/?art=ucboost&page=3
It has the button for start, the boost to 3.3V and the soft power off. All done with cheap parts since the MCU is used to keep the boost running.
How much power does a button push need?
Based on those, how big a battery does it need to be useful?
It looks like that the ESP eats up about 0.4Ws per button press, on my non-encrypted WiFi network connecting to a local server that is. If it's possible to completely drain the 180mAh battery that I use now, that should mean several thousand button presses can be done before the battery needs charging/changing.