And now back to the regularly scheduled programming! I had second thoughts about pursuing the inductor-based boost converter design to power a neon glow lamp with a standard 9V battery. Non-ideal effects are significant for both the inductor and the transistor switch and it wasn't obvious that I'd be able to get the booster to operate above a megahertz. I might return to this design later and hope to ultimately replace the inductor with a flyback transformer, but for now I'm going with a more straightforward voltage-multiplier boost converter.
For the first iteration, the square-wave oscillator is unchanged except that it's input voltage is doubled to 18V by using two 9V batteries in series. I wasn't sure how many stages the voltage multiplier would need, but with 18V input I got away with using a 4-stage Greinacher (a.k.a. Cockcroft-Walton) multiplier. Here it is on two mini breadboards (voltage multiplier to the left and NE555 square-wave oscillator in the back to the right:
The components were what I had available. The diodes were bargain "germanium" diodes I bought a couple of years ago. I'm pretty sure they're just generic silicon signal diodes, but they seem to work well enough for this application. The voltage rating of the caps was a bit marginal and I used the lowest capacitance values a SPICE simulation told me I could get away with without reducing the output voltage. I did this over two months ago and I don't remember the oscillator frequency, but it must have been less than 200kHz. Anyway, this first crude attempt was encouraging enough. Here's a cheap multimeter showing an output voltage of 80.8V, more than enough to light a neon glow lamp: