I found a website that described DC-DC converter basics. I took a crack at building a "boost" style step-up DC-DC converter (where an inductor is placed in series with a power source and the output side is switched to ground.) I managed to step 5 volts to 24 volts across a 1.2K load (using low-power components) for a current of 20mA or almost 0.5 watts.
I found that transformers work particularly well in the circuit — plus, the secondaries offer useful voltages as well. By switching to better transistors (i.e. 2N2222 instead of 2N4123) I achieved 38V out into 1.2K: 32mA or 1.2W. This looks very promising … now if only I could get it to work from 1 volt.
I measured across 51 ohms and got up to 9.3 volts out. The input current is about 0.46A at 5.7V, so that's 2.62 watts and the output into 51 ohms is 0.18A or 1.70 watts out, so it's about 65% efficient. Using a smaller torroidal inductor, I got 8.62V into 51 ohms or 1.47W with 5.82V at 0.40A in or 2.34 watts for 62% efficiency.
I started building one to work off 1.5 volts or so. At first I didn't get it to work. I rebuilt the whole circuit and got exactly the same bizarre result: a short-cycle square wave that seems to ring down. I couldn't get the thing to work. The capacitor on the NPN transistor seems to be running into negative voltage territory somehow … it actually oscillates, but the final output is a stilted square wave. I switched to a (possibly more stable) twin-T design which I managed to get to work with as little as 3 volts.
I thought that I could try using MOSFETs but I couldn't figure out how to get them to work.