I managed to get the ESP8266 to run a webserver for 90 seconds off 4.5F of supercaps. It's kind of fun to charge them, because I"m using my BooSTick. Now that I have a bunch of them around, I find myself using them all the time. You can check that project out by clicking here.
So now the question is, how fast can I charge several farads of capacitance off a current transformer? That's the next step.