If you remember the guy (and whatever happened to him, hasn't been seen here for years now), he'd probably get a kick out of this one.
Around 67% efficient, 1W capacity (the Chinese LED claims "3W", but a AA won't supply that much, at least not for very long). I forget what frequency, 100-200kHz I think? The three position switch provides two brightness settings, though there's also an "undocumented*" flasher mode, where you put your finger over the two pins, providing about 2M of leakage, enough to charge up the base bypass cap, and make it blink at about a Hz. (The resulting waveform has a pulse width of a couple microseconds, probably difficult to capture consistently on video, but more than bright enough by eye.)
*Until it became documented. Right here, just now.
I recall Watt Sun searched long to find a good core for the transformer... of course, it helps to look in the right places. This one's a powdered iron, mix #6 I think, so it's got plenty of energy storage, without getting hot. Solid ferrite is absolutely the wrong material, even for low power circuits; but if you can get something with a gap, ferrite is excellent. As it happens, you can easily calculate how many turns and how much gap you need for a core of given measurements; no need to guess. Powdered iron depends on the grade; #26 and #52 are almost better resistors than inductors, but the lower-mu grades (#1, #2, #8, etc), and anything MPP, Sendust, Kool-Mu, etc., are pretty good. For these, just measure the A_L (or look it up, if it shows up in the materials listings at Micrometals) and put on as many turns as needed.
The transistor is one of those low-Vce(sat) types. It'll do a couple amperes with less than 0.5V drop, and that with hFE > 50. They're pretty great, like MOSFETs with really high transconductance and an unusually leaky gate.
Tim