Why is solar power out of the question? A 2000 mAh 5V solar power bank costs $13 on ebay. You could run an ESP8266 indefinitely on that, assuming you get >8 hours of daylight.
That solar power bank weighs 300g and measures 15x15cm (6"x6"). Should be within the spec for most projects.
What Im using esp8266 for, is to control a strip of those ws2812b led lights, the chip will not need much energy, only sporadically to change light colors or on/off, so if I place that solar pad close to it - the led lights will charge the chip which controls them!
But what is powering the LEDs? Why can't that power source also feed the relatively tiny amount (compared to a bunch of LEDs) needed by the controller?
Right now the same power source, a simple adapter/transformer, is powering both. Thats the easy way, this solar pad way is just because it would be fun.
I'm working on this part and you'll definitely need to implement deep sleep, but also measure voltage. Right now I'm using ESP.getVcc() but im not 100% that is correct. Basically once the battery packs voltage drop below ~2.5v the user should get a warning it needs a recharge (just as a failsafe even when using the solar pack). Also, I haven't tried to see, but also make sure the battery pack allows the esp to draw current when in deep sleep or it wont wake up :$.
I'm not sure you need to measure voltage on one of these power banks; they're made to charge phones over micro USB, so they will have a DC-DC transformer giving 5V all the way down to empty, I believe.
Edit: also, I think the ESP8266 draws about 0.2W on average when it's on. If you have a 10 Wh battery, you can run it for 50 hours in theory. Would you then need deep sleep?
Bear in mind that solar cells are rated for the amount of power they provide in direct sunlight, and will generate much less under other lighting conditions.
The human eye has a huge dynamic range; you might not realize just how well it compensates for differing amounts of illumination. Many indoor environments are 100 or even 1,000 times dimmer than outdoor direct sunlight.
You can get a bigger one, sure, but it will obviously be heavier, bigger and more expensive. I think the 2000mAh ones are the smallest you can live with for this type of setup.
That solar power bank weighs 300g and measures 15x15cm (6"x6"). Should be within the spec for most projects.