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using a small 500V mosfet instead of the xtor would improve the efficacy a lot - 10 ohm gate resistor - it would need to be a logic level device if you plan to run off 5V - Make sure L3 cannot saturate at full peak current.
There is the flyback converter. It uses a transformer which provides isolation. Its duty cycle is easy to work with. The simulation below assumes the GM tube is 10 Mohms.
OTOH the boost converter steps up 100x by means of a very long duty cycle followed by a momentary spike of high voltage. The high voltage is generated at the lower terminal of your inductor. The possibility is that it might find a path to low voltage devices.
My geiger counter is Victoreen 700, vintage 1961, common in civil defense shelters. Discrete components, solid state. It boosts 6V to 900 V. The method chosen was a flyback converter driven by a blocking oscillator.
Schematic is 2/3 of the way down the linked webpage.
It's a flyback converter, the winding ratio can be smaller than voltage ratio. There are small off-the-shelf SMPS transformers that can be used, but they are of course still bigger than simple inductor. The switching losses are proportional to peak current multiply flyback voltage, thus much higher with the inductor solution, but possibly still acceptable due to the small output current.
Using a rectifier with voltage multiplication is useful in any case, consider that small storage inductors are often not rated for high voltage.
I would consider a current mode SMPS controller instead of 555 with auxiliary circuitry.
A single diode ( 2 in series for safety ) would be better for the output - given that device is 600V, rather than the 3 diodes and caps that appear to serve no purpose - in addition to the suggestions put forward above ...
Voltage multiplying rectifiers have generally a higher output impedance, caused by the series capacitors. The absolut value and ratio may be tuned for a specific application.
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