Theory behind the "Joule Ringer" circuit
Added at top #2:

This is beginning to look more like fraud with a small chance of ignorance.  

This video allegedly shows a version of his device driving a significant bank of light bulbs. It appears he is getting 10's of Watts out and he says input is 200 mA. Voltage is unspecified but if it's 12V in then power = 2.4 Watts. If so he has some awesomely efficient light sources.
He MAY be simply saying that he has developed a switching regulator that seems to work efficiently but the average person would take this as a "free energy" claim. He does not say so and nowhere provides enough plain technical data to pin down what he IS saying.

So: Either he is making high efficiency inverters - there will be better ones, or he is claiming net energy production and is a fraudster (or he is working magic). .



2nd oldest below here:

Added at top #1 :

I hadn't noticed the high current and inductance posited to get the 25J suggested and I did no more than skim the material available. That current is possible but I'd need to take more time investigating it at present than I can spare on what is either a hoax (probably not) or a misunderstanding.

I had not realised that this is a claimed alternative energy magic power source concept. 
It is never worth my investigating those as the magic goes away and hides as soon as I come near and refuses to come out to play until after I leave. Alas.

If this is able to be replicated it is able to be measured and the people doing it should provide measurement results. If the system uses magic which cannot be measured they should say so and we can get on with other things. 

The videos &amp; text I noted seemed to be talking about operating times in the 30-60 minute range. Somebody else mentioned 12 hours but you'd want a signed affidavit on that one.
You can get a good appearance on an LED with 1 mA and a bright flash at an average of say 0.1 mA. At 1 mA and say 2.5V you need 2.5 mJ/second, 50 mJ/minute and 3000 mJ or 3J or 3 Watt seconds for an hour. A 2000 uF capacitor charged to 10V contains 100 mJ of energy (not all easily used) or about 2 minutes of LED operation at 1 mA mean. Inductor energy my explain what is happening but you'd need relatively immense currents and substantial inductance - more of both than seem likely to be present here.

One other possibility is "energy harvesting" from RF pickup. Seems unlikely.

So  - hat I wrote before I'll leave BUT in fact I think either magic is happening (my job here is done) or things are misunderstood, or a series of pulses convey a sense of brightness at actual low power or its a hoax. The experimenter SEEMS genuine enough - but this would be very easy to 'cook' and I have seen any number of fraudulent claims. 



Oldest below here:

There is no magic here, alas.  I see nothing that surprises me energy wise.
A good modern LED will provide useful output at 1 mA so your energy calculations are in the right order of magnitude for what you are seeing.  

A look at related circuits and associated videos show that the actual circuit in a given case may not be exactly what is drawn and also that they are using parasitic coupling of various sorts in some cases.

I will come back later and add a bit to this but for now here are some useful related links.

V2 - hard connection to base shown for feedback circuit

V3 - high value resistor provided for feedback

Another version = also has video