The schematic I posted shows a circuit containing 2 diodes (D2, D3) and 2 capacitors (C4, C5) on the secondary side of the step-up transformer T1. This configuration is known as a voltage doubler. By adding additional diodes and caps, you can achieve even higher degrees of voltage multiplication. Such circuits are called Cockroft-Walton multipliers, and are commonly used to generate very high voltages for everything from bug zappers and CRT type TV sets to huge particle accelerators and X-ray machines.
https://home.earthlink.net/~jimlux/hv/cw1.htm
A battery powered photoflash works by chopping the DC from the battery into pulsating AC, and then driving a small stepup transformer to generate 2-300 V for the flashtube. The Geiger counter schematic I posted works the exact same way. The 4049 inverter is set up as an oscillator (U1A and U1B), which feeds an inverting buffer (U1 C+D) which drives the primary of the T1 transformer through the MOSFET Q1.
To modify a photoflash, I would disconnect the circuitry (rectifier, capacitor, "ready" light, trigger coil, etc.) from the secondary side of the DC-DC converter transformer, and use the transformer secondary to supply AC to a 3-stage multiplier. A photoflash may contain a lot of other circuitry that you won't need/want, so you might want to just salvage the tiny stepup transformer from the flash, and use it to build something like the posted circuit.
A 3 stage voltage tripler on the output side of a photoflash transformer should get you somewhere in the 1kV range without a problem. You actually want the no-load voltage to be higher than the tube will require, because you will be regulating it down using a shunt string of zener diodes.
If you end up using a photoflash as the basis of your power supply, you need to make sure to REMOVE the main energy storage capacitor (discharging it carefully before handling, of course). A Geiger tube does NOT need a power supply that stores a lot of energy, and leaving the photoflash capacitor in creates a needless hazard. The capacitors in the tripler circuit will provide sufficient filtering for this application.