Like a lot of experimenters, I started with a neon sign transformer as a high voltage source. Common neon sign transformers cost about $80 and put out about 15kV (no load) and around 30 mA shorted, although you can get lower voltages and currents easily. You can also get bigger transformers, however, they are special order, and are more expensive. Furthermore, they have more current output, but the voltages stay in the same 15 kV range. In the neon sign business, there just isn't much call for higher voltages. Also, the standard wire that signs are wired with is rated to 15 kV, providing a nice 2:1 safety margin, since the center of the transformer secondary is grounded (so it really puts out only 7.5 kV relative to ground).
A neon sign transformer limits the current by its construction with a high leakage inductance, which essentially puts an inductor in series with the transformer. The impedance of the series inductor serves to limit the current, while not dissipating much power (as a resistor would). This is desirable when running a discharge tube, because the tube has a negative V/I characteristic, and if driven from a stiff voltage source, it would draw more and more current until either the tube or the transformer melted down. The same technique for current limiting is used in arc welders, although at a somewhat lower voltage and a lot more current.
However, we all want more voltage or current. Let's look at voltage first. You could just increase the primary voltage, which will raise the secondary voltage in proportion. There are two basic limits to this approach. The first is that the insulation on the transformer isn't necessarily good to much over the nominal voltage. They have already gone to the expedient of grounding the centertap to reduce the voltage requirement. You might get to 20-25 kV before you started having problems. The second problem is core saturation. They don't design transformers with a lot of extra iron, because it is expensive and heavy. As you increase the voltage, you need more area in the core to keep the flux density below the saturation level for the iron. If the core saturates, you draw more power, but it essentially all goes into heat. You can see this if you hook a 0-240V autotransformer up to just about any 110V transformer, and gradually increase the voltage while monitoring the output voltage at no load. Around 150-160V, the output voltage stops rising proportionately, and if you look at the waveform on a scope, it becomes quite distorted.
Now let's look at current. What limits the current is that series inductance. In theory, you can put in a capacitor of equivalent impedance (although opposite in sign, of course), which will allow more current to flow. The problem with this has to do with cost and availability. Say your neon transformer is rated at something like 400 VA, i.e. it draws a short circuit current of around 4 Amps. The equivalent series reactance providing the current limiting is about 25 ohms, or about 60 mH. A capacitor to just cancel this would also be 25 Ohms, or about 100 uF. The real problem, though, is that the transformer is designed to run at a particular power, and if you increase the current, you increase the heating from the I2R losses and core losses. Draw twice the current, and you have 4 times the heating, which will probably melt the insulation.
Well, what about hooking up multiple transformers in series or parallel? You can hook up multiple transformers in parallel to increase the current, although at some point, it is cheaper to just get a bigger transformer. If you are buying new, this is certainly true: a transformer of twice the rating costs less than twice as much, and is more efficient to boot. However, if you are buying surplus (typically 10%-20% of new price), you have to work with what you have got. When hooking up parallel systems, make sure that parallel outputs are properly phased. There are also some potential problems if the transformers are not exactly identical, particularly with respect to series impedance (they will probably all have the same turns ratio). There are some tricks using an additional smaller transformer to balance multiple paralleled transformers.
<figure here>Theory behind this scheme
What about hooking up transformers in series to get more voltage. The problem here goes back to insulation and grounding. A neon transformer has the secondary centertap grounded to the case, so the insulation from the secondary winding high voltage end needs only withstand the 7.5 kV. If you remove this ground, you might arc to the case (not so bad, as long as the case is isolated) or much, much worse, arc to the primary side (actually more likely). Then, you are feeding kilovolts back into your power line, a bad situation in any case.
If you have a cheap source for isolation transformers that can take many kV, you could run one
transformer grounded, then run the other through the isolation transfomer, and successfully series
them. Isolation transformers like this turn up in surplus sometimes, they are used to power 110V
lights and such that float at high voltage (like tower lights on transmitting antennas) or to run small
motors or relays which are sitting at high voltage line potential. Unfortunately, they are typically
around 500-1000 VA, and so are not suitable for high power systems (unless you have a lot of
bucks, or the transformers are very cheap).
There is a solution, however. Assume a transformer with 3 windings: the 110V primary, the HV secondary, and another 110V tertiary. Also assume that the insulation of the transformer is such that the two 110V windings can take the full secondary voltage. If this is the case, then you can feed a second transformer from the first, essentially using the first transformer as a isolation transformer. The primary of the second is fed from the tertiary of the first. You can, of course, extend this ad infinitum, feeding a third transformer from the tertiary winding of the second, etc.
The limitation is that the primary of the first transformer must carry the entire power of the chain of transformers, so those primary and tertiary windings have to be pretty good sized. For instance, if you had a chain of 10 transformers, each putting out 10 kV at 100 mA (i.e. 1 kVA), the primary of the first one in the chain has to handle 100 Amps: 10 Amps for the power to the HV secondary, and 90 Amps to the tertiary winding. The second stage handles 90 Amps: 10 to the HV secondary of this stage, and 80 to the next, and so on.
It so happens that C&H Sales sells the aforementioned illumination transformer which conveniently has a tertiary winding. The basic specs of the transformer are: 110V primary at 8 Amps, 4500 V secondary @ 450 mA, and a tertiary winding at 600 V. This is a transformer designed to run a discharge tube, so it was designed with leakage inductance to limit the current: you can short the output and the transformer doesn't overheat or overload. The 600 V winding is designed to connect to a 10 uF cap, which provides a measure of power factor correction so that the power line doesn't see the highly inductive nature of the transformer. The designers did this because 10 uF, 600V caps are readily available and cheap, as opposed to the 3600uF 110V cap that would have been necessary if you just tried to do the PF correction with a parallel cap.
So, we can hook up a second one of these transformers by running the 600 Volts out of the first one into the 600 Volts of the second one. We leave the 110V primary of the second transformer unconnected and connect the 4500V secondaries in series. The second transformer core is connected to the low side of its secondary so that it runs at some known voltage, as opposed to whatever capacitive coupling would get it to. And, of course, it has to be mounted on insulating standoffs of some sort. Now we have a 9000 Volt 225 mA transformer. The leakage inductance of the first transformer of the chain limits the current into it, which effectively is divided between the two.
We can hook up another pair of these transformers, phased exactly opposite, with the primaries in parallel, to get a 18 kV center-tap grounded transformer at 225 mA, or, an 18kV, 225 mA neon transformer. It draws about 16 Amps with the output shorted.
This unit, although a bit heavy, produces very nice fat sparks in jacob's ladder type systems. The diameter of an arc is determined by the square root of the current (i.e. the current density of an arc is roughly constant), so the 10 fold improvement in current gives you an arc some 3-4 times larger than the typical 30mA neon transformer. All this for about $200 in transformers, and some insulating standoffs and chassis work. I should mention that the chassis work is pretty important, since you are going to wind up with about 200 pounds of transformers. I built mine on a sheet of aluminum that was bolted to a hand truck with wheels. You could also build it inside a milk crate or something like that.