Anyone ever build a stack capacitor out of paper and cut up beer cans?

You can reasonably expect to achieve a breakdown voltage of several hundred volts. Possibly as much as 800 to 1000V but more likely 500 to 600V. Paper is an insulator and the breakdown voltage will be dependent upon the quality and thickness of the paper as well as its moisture content. You would be wise to dry the paper in an oven. Once assembled you can seal the cap in wax to keep moisture out. In the early days of electronics capacitors were made using paper and were sealed in wax.

Plastic would make a better dielectric material, it has a higher breakdown voltage and it can be much thinner than paper which would yield a higher capacitance. Instead of beer cans you can use aluminum foil, it easier to work with and smooth out.

Spark gap radios used stacked capacitors which used sheets of glass as dielectric material.

EDIT:

In the link below you will find pictures of a vintage spark gap transmitter that is fully functional. Notice in one of the pictures you will see a blue warning sign "Danger Keep Off Live wires " Just below this sign is a cabinet that contains a stacked plate capacitor which uses glass plates.

Edit:

Piercy makes a point about Series configuration of stacked pates. This will distribute the voltage across the plates as he describes but at the expense of lowering capacitance as the net result is an cumulative increase in dielectric thickness.

Stacked plates can also be electrically connected in parallel to achieve higher capacitance. Doing so increases the cumulative surface area of the plates while keeping the dielectric at its minimum thickness. This results in an increase in capacitance and the total voltage applied across what are effectively two plates.

This is done in the stacked cap for the transmitter. Notice heavy green wire connected to a shorting bar. The bar it attached to every other plate in the stack. There is another on the opposite side which interconnects the remaining plates.

Put it another way. Stacked plate refers to the physical stacking of the plates and dielectric and not the electrical connection of the plates.

EDIT:

Percy also mentions charging a AC using line voltage which is 120V AC (RMS). To reliably charge a cap you use DC by rectifying AC using one or more diodes. The peak voltage of the sine wave is 1.414 times that or nearly 170V. So your rectified DC voltage will be higher than your 120Vrms AC.

You may even use transformers to boost the AC voltage before rectification into DC. Just be aware that the cap will charge to peak voltage which will be 1.4 times higher than what your RMS AC voltage is. DVM's readout AC volts in average or RMS NOT peak so take this into account when you design your transformer and CAP......

To effectively and reliably store a charge on a cap you use DC.

I have never performed a di-electric strength test on photocopy or computer printer paper but it is sure to be several hundred volts. Since you are making a stack capacitor, each piece of paper will be in series. Given this fact, if the capacitor consists of 20 pieces of paper, and the applied voltage is 120 Volts, the voltage drop across each piece of paper, will be 6 Volts. If you are intending to test your capacitor by connecting it to the public supply, you should do so under proper supervision. Furthermore, once the capacitor is charged by a 120 Volt supply, it is the same as a 120 Volt battery and therefore is very dangerous --- DO NOT TOUCH THE TWO ENDS OF THE CAPACITOR.

Yes, paper cut.