I should have given some more details. The battery is a 3.7V 3000mAh lithium ion cell. Only at 5V, we need a 1A supply, which is possible with the efficiencies that switching regulators have. At 12V, we are looking at lower supply currents (around 0.5-0.65A).
You can increase output voltage to around 200 to 300 volts if you introduce a ferrite core in your inductor (at the max for the frequency of your timer). You can use a cheap TIP162 or TIP3055 darlington transistor and they do not heat up even with continuous currents of 10 Amps and peak currents of 15 Amps. It is pretty efficient for a boost converter and if you increase the frequency of the timer to 100,000 Hz and use a fast recovery diode of the RUR.... series you can get the voltage to around 50 Kv But you need a fast recovery NPN power transistor for that.