OK, so I made some progress:
Realized that I had some poor connections at the 555 killing the pulse to the MOSFET. The timer is now working fine.
Once I got it all hooked up I used an IRF740 MOSFET as the switch. I made an adjustment to the schematic and added a variable resistor at both R1 and R2 to play with the duty cycle. I had to use a pretty heavy duty toroid inductor I made to get any decent charge build up. This baffles me as I have seen other circuits use smaller inductors with better efficiency. One circuit design I found used a small 200 uH coil to achieve 300 volts. The timer is remains cool but the MOSFET gets hot. Also, if the coil is big enough, I get a huge voltage spike according to my voltmeter as it will read something like 1700 volts but when I kill the circuit the cap array is only at 120.
So, to remedy the hot MOSFET, I upgraded to an IRFP450 but it does switch on. I thought it was bad then tried another one but nada. The circuit works with the IRF740 though I can't find an adequate duty cycle that charges to 300 volts in a stable manner. I have compared my circuit to other designs and I can't see what is wrong. According to other circuits I have found online, this should be working yet isn't.
I'm at a loss here as I calculated that I need a 95% duty cycle for a small inductor and from there I got the resistor values. Yet, they didn't work and especially with the recommended MOSFET (IRFP740).