I have some wall warts that step down 120V~ to 5V~ and rectify it. This is a turns ratio of 24:1. Could I, in theory, flip around the transformer w/ out rectifying and get about 2.88kV? (120*24=2880) I realize that amps would be very low.
If it has an AC output you could run it theoretically 'backwards', however not at the voltage levels you speculate on because of the voltage rating of the insulated wire used in the internal transformer windings. If the rectification is internal to the wall wart then it has DC output and cannot be used 'backwards', as DC will not pass through a transformer.
Could I, in theory, flip around the transformer w/ out rectifying and get about 2.88kV?
In theory, yes - but you probably wouldn't be able to pull a lot of current. The coils of a transformer are made specific to the task at hand - the number of turns, the gauge of wire, etc - all play into the specifics of what the transformer is designed to do.
With that said, you can still have a bit of fun flipping a "standard" transformer "backwards" (note by this I mean a "bare" transformer; the component inside the wal-wart). Back when I was a kid, I used to make relay "buzzers" to drive multi-coil TV transformers (which you really don't see any more, since TVs typically use some form of switching power supply instead of a linear supply like the old sets did) backwards for "shocking" devices.
We (me and a few friends of the same bent would make these) used to take them to school, and have kids hold the ends of the wire, and rather than a relay, we would touch the wire briefly to our battery (usually a 6V lantern battery), to "pulse" the transformer - and see who could hold on.
At one point, with my buzzer circuit, I was able to use some carbon rods from old carbon-zinc cells and jump a spark between them.
I tend to think it was a good thing that as a kid, I knew nothing about voltage doubler circuits, or how capacitors actually worked, etc; likely would've seriously hurt myself or others.
Keep that in mind - high voltage is not something to take lightly; it is something to be well respected, because if you aren't paying very close attention to what you're doing, it may be the last time you do anything!
The design of a transformer entails ensuring there is sufficient iron core top prevent magnetic saturation. This is based on so many square millimetres per watt of power transformed. The mains side of a transformer has sufficient turns to create sufficient impedance (inductive reactance, not DC resistance) and and this is based upon so many turns per volt applied (based on having an unsaturated core)
If you reverse primary and secondary and try applying mains to a low voltage secondary, you have sufficient iron for the wattage to be transmitted - that hasn't changed, but you have insufficient number of turns for the voltage being applied.
Reversal is often done with 230 to 110 transformers that are specifically designed to work either way, but they are wound to ensure there are sufficient number of turns, whichever side is chosen as the primary.
All above are correct.
The core saturation limit can be avoided, at least fer a while, by increasing the frequency. But insulation breakdown will still gitcha, probably a LONG way before 2.88 kV.
If your low winding is center tapped you can use a 2n3055 to make a power oscillator to step up a battery to a few hundred volts. Makes a dandy cattle prod, or will light a fluorescent tube.