TinMan asked a good question, could the whole AA battery be wound on and work in the Joule Thief circuit.
Yep, works well, the whole battery is covered.
The connections are - the end of 1 wire connection to the start of the other for the Positive connection. Thin wire to the Base of the transistor, thicker wire to the Collector.
Between the coil layers is a layer of packing tape, to act as a dielectric and to protect the first windings from those above.
Current draw for a 1W LED is 50mA and the output is approx 1/2 full brightness.