I thought I would use a relatively new AA battery to power a flickering Joule Thief to see how its battery voltage varied with time. The image here shows the resulting voltage versus time relationship.
The ACDelco alkaline AA battery was bought a couple of years ago from Fry's Electronics. It was unused initially (if a little old) and ran the Joule Thief for 6.75 days. I must admit that I was disappointed by this performance - I had been expecting more than 7 days. (I am not sure why I expected more, the device draws 38 milliamps from a fresh battery. A typical alkaline AA battery is rated at 2000 milliamp hours, so should only power the device for 2000/38 or approximately 50 hours). Anyway, I was optimisitic, because I figured that as the batteries voltage went down, so the current drain would drop. To some extent this was true, because 6.75 days is much more than 2 days (!) but the Joule Thief did not run down the battery as far as I thought it would in voltage terms. The LEDs stopped flickering somewhere between 0.67 and 0.63 volts, and I had been expecting something more like 0.45 volts.
This particular Joule Thief circuit (as shown here) has voltage stabilization and this is probably costing quite a few wasted milliamps. The coil is also likely not the most efficient - I used a ferite rod rather than a torroidal core. Taking a battery that can hardly light the flickering LEDs in this circuit and using it with a standard Joule Thief shows the expected behavior with the LEDs still being powered down to about 0.48 volts.
You can make semiconductors which operate at lower voltages and so can extract energy from batteries down to very lower voltage. For example, germanium transistors work down to about 0.2 volts and JFETs and CMOS devices even lower voltages I believe.
So this device could be improved, but it does get something extra from with otherwise useless batteries - and I quite like it.