The type of circuit you're describing is called a switched capacitor charge pump. Switching circuits aside, it may not be suitable for what you want to do, and here's why.
Take a 2200 µF, 25V electrolytic capacitor that has been charged up to 12V. Now start drawing a current of 10A from it. The voltage across the capacitor will start to drop, as it discharges.
The formula for this dropping voltage is:
dV/dT = I / C
where dV is the change in voltage across the capacitor, in volts;
dT is the time period over which that voltage change occurs;
I is the current being drawn from the capacitor, in amps, and
C is the capacitance, in farads.
Plugging in the values for I (10) and C (0.0022) gives a rate of voltage change of 4,545 volts per second, or 4.5 volts per millisecond.
In other words, after just one millisecond (1/1000th of a second), the voltage on the capacitor will have dropped by 4.5V.
For the output smoothing capacitor, if you want less than, say, 0.45V peak-to-peak of ripple on the output voltage, you will need to recharge it 10,000 times per second.
Ripple will also occur due to the capacitor's internal resistance or ESR (effective series resistance) but this is less of a contributor, as long as you use low-ESR capacitors.
That might be feasible I suppose, but you also need to consider the ripple current specification of the capacitor as well, which is around 1~2A RMS for a good quality low-ESR 2200 µF electrolytic. Ripple current causes heating and reduces the lifetime of electrolytics, so it's best to be very conservative. Paralleling around ten electrolytics will force them to share the ripple current. You should still use good quality Japanese electrolytics.
Increasing capacitances will also allow you to use a lower switching frequency, which reduces losses due to gate capacitance in the MOSFETs.
The other capacitors in the circuit will also need similarly large capacitances and ripple current ratings.
You should definitely use MOSFETs instead of transistors. There are many differences, most of which make MOSFETs more suitable for this project. The most important is the ON characteristics. A saturated MOSFET behaves like a resistor, and it can have a very low resistance - just milliohms, for the big grunty ones! A transistor, on the other hand, has a minimum Vce(sat) voltage (which is much higher for a Darlington), which guarantees significant voltage loss in the transistor.
An N-channel MOSFET can be compared to an NPN transistor, and a P-channel MOSFET can be compared to a PNP transistor. So you can choose the type of MOSFET according to the circuit position, to make the drive circuitry easier. But driving MOSFETs is quite different from driving bipolar transistors.
Get some application notes on MOSFETs from companies that make them - go to
http://www.digikey.com/product-search/en/discrete-semiconductor-products/fets-single and look through the manufacturers, go to their web sites, and look for design support and application information on MOSFETs in general, and applications similar to yours.
A charge pump voltage booster that generates nominally twice the input voltage needs only one switched capacitor. On one half of the cycle, the capacitor is charged from the input voltage, and on the other half, it is added to the input voltage and discharged into the output smoothing capacitor. You can connect a large capacitance across the input voltage as well, but that capacitor doesn't need to be switched.
Sorry for the disorganised nature of this reply :-(
Please keep us informed and post your design for us to check.
