I was wondering what process, or processes, would yield the highest caffeine quantity at a fixed drink size (Say 200ml).
Assuming the same variety of bean, same roast, same quantity, same water temperature and same extraction time.
EDIT: I thought about this more. In sum, I believe that this question (as asked) is not answerable (but it is interesting!). With just about any selected brewing method, you'll be able to extract almost all of the caffeine from the beans. However, depending on the method chosen, you need to use different parameters to get full extraction (time, pressure, temperature, etc.). For example: Espresso is certainly the fastest to get full caffeine extraction, but you need a fine grind. Filter/drip or French press are slower, but espresso-fine grind just won't work correctly; you need to use a coarser grind. Given the same beans, and the right technique for the given method, you can achieve almost 100% caffeine extraction.
As an example of drip versus espresso, the Wikipedia article on caffeine (and the two cited references that include figures on both espresso and drip caffeine content -- this and this) claim nearly the same caffeine content for drip versus espresso; in fact, they list that drip coffee has slightly more caffeine than espresso. I can't fully accept these, however, because there's no way to know if they're using arabica or robusta, light roast or dark roast, how much ground coffee was used to brew -- there's just too many unknowns that back those figures. Since arabica, for instance, is ~1% caffeine, with 7g of coffee ("standard dose", from INEI) you'll be able to get ~70mg caffeine. Those are pretty consistent with the cited figures -- meaning 100% caffeine extraction.
So, you'd have to ask a different question in a different way (e.g., what's the fastest extraction method? or How to get 100% caffeine extraction with cold brew?) or so.
I agree with most of @Chris in AK's answer (and +1!), but I wanted to add another take on this.
I don't think solubility is a big factor here; caffeine is very soluble in hot water -- approaching 66g/100mL (~150g/cup) in boiling water, and even at room temperature nearly 2g/100mL (~5g/cup) (citation listed at Wikipedia and here). Yes, those are listed as grams, which means that water could dissolve tens to hundreds of times as much caffeine, as compared to the caffeine dose in an average cup/demitasse of strong coffee or espresso (which contain perhaps ~50mg-200mg per cup; see also questions on caffeine content, such as this and this; also keep in mind that the lethal dose of caffeine is somewhere in the 10g-20g range.)
Back to your actual question, I do agree that you've restricted yourself quite a bit; pressure is one of the few variables you didn't eliminate. To that end: a comparison of extraction of pressure-brewed (e.g., espresso) versus filter/drip follows.
In general, caffeine is quickly extracted from coffee beans (and tea leaves); most is extracted within 30 seconds of water contact (though this certainly depends on fineness of grind and other factors). Here's a visualization of extraction of comparison of pressure-brewed and drip versus time, which depicts the quantity of caffeine (and other stuff) extracted over time. I found it amusing that the context of that article is actually advocating for espresso as a way to brew lower caffeine content coffee; the point being that the "other stuff" (aroma, colour, flavour) is extracted even more quickly with pressure-brewed (versus drip) coffee. So, if you extract for only 10 seconds (which is very short indeed) you'll get most of the "other stuff" but less caffeine. On the other hand, a finer grind will permit faster extraction of caffeine and everything else; this article suggests that espresso is higher in caffeine based on the grind factor; I think there are too many other variables for that conclusion.
I think the answer to your question is generally going to be espresso.
Brewing process all comes down to pressure, temperature, extraction time (water contact time), possibly agitation and filtering. I may be missing a variable, but all the methods of brewing are just mixing of those variables. If you fix all but one variable, you'd have an answer for any given equation.
All of these things are ultimately affecting the solubility of the caffeine in the coffee. Since espresso is one of the few brewing methods to affect pressure (and generally uses the greatest pressure) it's going to be the winner. Increased pressure means increased caffeine solubility.
From wikipedia you'll note the two major factors affecting solubility are temperature and pressure. You'll also note that dissolution doesn't always happen instantly; rates may vary based on conditions. By increasing pressure during brewing my guess is that the rate of caffeine dissolution is increased. making it more efficient for short brewing times.
This study on the caffeine concentration of Turkish Coffee, shows that it can have concentrations of up to 300 mg per 40ml cup (table 6). This method of preparation has much stronger concentrations than the espresso or the drip coffee, since you end up drinking a lot more of the grinded coffee beans.