Paul:
These things are indeed fun! Another interesting source of radioactivity is “uranium glass” or “vaseline glass”, which is yellow fluorescent due to added uranium and thus weakly radioactive. There are also the famous radioactive orange glazed ceramics that were popular a couple of decades ago. For a very hot source of alpha radiation, you can take apart a smoke detector to get at the polonium emitter. Keep in mind that it is extremely dangerous to inhale or ingest polononium, so everything should be handled while wearing gloves and a good air filter mask.
In the circuits above the HV supply is unregulated, so the output depends on the average current draw by the Geiger tube. For the LND7313 tube, the pulse is about 200 usec long, with a peak current of about 30 uA. At 1000 counts/sec (a pretty hot source), I calculate that this corresponds to an average current draw of 3 uA or a drop of 30V across the 10 Mohm resistor. That is acceptable.
In my vicinity the average background is about 1 count/second, but you should see how that shoots up when you go up over 5,000 feet elevation! Cosmic radiation at 30,000 feet is about 100 times as high as at sea level.
For a PWM HV source example, Nuts & Volts had an article a while ago showing how to make an LED flashlight with a 100 uH inductor and an ATtiny13 running 13 bytes of code:
nutsvolts.texterity.com/nutsvolts/200802/?pg=33
It would be easy to adapt this to power a Geiger counter.
Jim