I know it seems like a crazy and useless idea but I get enjoyment out of doing things the wrong way. With that being said, I wanna have a small series circuit just for fun.  I've heard of people calculating the voltage needed to run it by dividing the wattage by 6.6 amps.  But I fear that this equation does not account for ballast losses. So does that work out, or is there another way of calculating this? I was thinking I could have four ballasts, either 200 W sodium or 175 W Mercury in series and it would work.  Now it would draw tons of current, but it would be fun just for the sake of its existence. What am I missing here?

With incandescents, the "Divide the power by 6.6A" would work, with or even without the losses (they are not that big).
But with discharge there is one huge problem: Discharge requires a series impedance, so it maintain the current even when the lamp arc voltage varies. That variations are general arc voltage fluctuations during warmup or lamp aging, as well as the shape miosmatch between sinewave mains and nearly a rectangular shape of the arc voltage waveform, plus the need to deliver elevated voltage (above the normal arc voltage) each time the current crosses zero to reignite the arc for the opposite current direction (= 120x per second). Therefore any discharge needs to be fed from a rather high impedance source, so a power source which does maintain the current even when the voltage goes crazy (when exaggerating only a little bit). Normally that is the main function of the ballast designed to operate on a low impedance source like the mains.
In the series street lighting circuits the constant current regulating transformer feeding the branch is, what in fact performs the ballasting function, it uses to be a transformer with higher OCV than the circuit will operate and by its leakage inductance limits the current (the leakage inductance is adjusted by the constant current mechanism to reach the exact 6.6A output).

The series circuit "ballasts" are then not ballasts at all, just (power current) transformers converting the 6.6A into whatever current the lamp needs, as the branch feed transformer is what performs all the ballasting. And being just transformers means, they just translate the circuit impedance by a fixed ratio. So if connected to very low impedance mains, the secondary will exhibit low impedance output as well, so not suitable for any discharge. Connecting multiple discharges in series won't change anything, still the total chain will need  the OCV at least 2x the total arc voltage, albeit all transformed to the levels corresponding to the 6.6A rating.
Incandescents will work, because incandescents don't care whether fed from a voltage or current source, so if the sum of all the voltages matches the mains voltage, it will work fine (assume not that wide tolerance of the lamp parameters).

So in order to make your "6.6A system", you need to build it with all the components, including the ballasting feed transformer. But for the "feed transformer" you may use an MV or probe-MH (= no ignitor) ballast for a lamp rated at 6.6A arc current (I would guess somewhere along the 750..1000W lines, you have to really check the nominal currents). Without this it would not work.

So let's say I took a 1000 W ballast and somehow figured out the right value capacitor to make it so that it puts out 6.6 amps of current, regardless of the output voltage, you're saying that the 6.6a "ballast" cans would theoretically work fine off of it?  Because I have a 600 watt sodium ballast that might work pretty well for this.  And I'm assuming I would have to watch the overall wattage of the ballast's chained together so that it equalled close to what the ballast was rated for?

Indeed, selecting the capacitor so the load current becomes the 6.6A should work, but never go above the original rated current of the ballast (I don't remember the 1kW arc current rating).
The maximum  total power should keep the load voltage below half of the ballast OCV, but if the real power does not reach and/or exceeds the "rated" power is not relevant that much (I mean +30/-50%). Relevant is the current, the 6.6A should not be even a bit above the nominal arc current rating.

So I have a 200 W high pressure sodium ballast meant for a series setup.  It's rated for 6.6 amps, but is that how much it draws or does it stand for something else? If that is the case, could I just put it on said 600w ballast that puts out that ~6a ?  Or do I absolutely have to put a few together to match the wattage?

The series stuff is rated for a given constant current feed. So the power source dictates the current (e.g. the 6.6A) and that is what has to match, the voltage is then what the load drops when drawing the given power, so for that 200W it would be about 30V drop.

What I meant "don't bother by the power" is, when you use a "600W" ballast to become the 6.6A constant current source, the fact you are drawing from it just 200W won't matter, as long as the current stays the 6.6A.
But as the 6.6A current source (assume you are tweaking the xapacitance to get the current exact) you may use only a ballast that is designed to feed current at least that 6.6A. You may be able to persuade some smaller ballast to deliver the 6.6A electrically, but the winding will be overloaded so overheat. If you stary from some higher current ballast (e.g. 7.5A arc current) and reduce the capacitance so the resulting current gets the 6.6A, it is perfectly OK as there will be even less heat dissipated than designed.

So it does work, tested it with a 175 W Mercury vapor lamp because I couldn't find the ignitor I needed for the sodium one. I know it's not the best for it but just for experiments.  I'm gonna have to monitor the voltage on it. Waiting on another ballast in the mail soon. Thank you again.

I do have another question, if I have 2 of those series Transformers on the 600 W sodium ballast, wouldn't they be pulling the 13.2 amps as a total instead of 6.6?

So it does work, tested it with a 175 W Mercury vapor lamp because I couldn't find the ignitor I needed for the sodium one. I know it's not the best for it but just for experiments.  I'm gonna have to monitor the voltage on it. Waiting on another ballast in the mail soon. Thank you again.

Is that ballast loud?

The Series ballast isn't, the main one buzzes

I do have another question, if I have 2 of those series Transformers on the 600 W sodium ballast, wouldn't they be pulling the 13.2 amps as a total instead of 6.6?

No. In a current fed series circuit the current stays tge same, just the total voltsge drop will increase, as there are two loads in series.
The "feed" ballast would actually likely feed a bit less current, because the load voltage would get higher and it is not a perfect constant current source, but when talking about 20 or 40V load drop on a 200+V OCV, the difference would be very small.

OK so sort of like putting batteries in series but the opposite concept. Thank you once again.