This simple "Joule Thief" self-oscillating LED circuit will produce light even when the battery voltage drops to 0.4 V. Of course the light will dim as the voltage drops, since there is no current regulation. The design is based on an example circuit from Wikipedia: https://en.wikipedia.org/wiki/Joule_thief
For the transformer physical design, the user can explore different magnetic toroid core sizes, material types, number of winding turns, etc., to see their impact on the LED performance. This "virtual design" test-bench is particularly useful to view core saturation and its current limiting affect in this particular circuit. For example, try using a smaller core with 1/5th the area and length of the original.