What method of coffee preparation gives the most caffeine in the finished cup?
This question is nearly unanswerable. Typical caffeine content data is given as a range based on the normal brewing factors of each method, there isn't data for abnormal methods i.e. french press with fine grind or espresso with coarse grind.

Typical caffeine contents:


3oz Espresso double shot: 70-200mg  
8oz drip: 60-100mg
6oz French Press: 80-100mg
6oz Aeropress: 60-80mg


On average the espresso or moka pot coffee would have significantly higher caffeine concentration than other methods (with the moka pot maybe edging out the espresso because it uses higher temperature for extraction). There is so much variability that I don't know if there's any point in trying to answer this question. The article linked below shows that one method alone (espresso) can yield shots with caffeine concentrations that are radically different:


  The extended range of caffeine values found in retail coffees in this study suggests that it is presently not possible for individuals interested in assessing the caffeine content of their diet to do so if the intake includes retail coffee. Our data highlights that, if just two retail coffees are consumed in a day by the same individual, that a variance of more than 290 mg can occur if both of these espressos are purchased from either the highest or lowest 10th percentiles of all the beverages sampled


If you control for volume by diluting everything with a typical serving size less than 200mL, all the methods are going to yield caffeine doses in the range of 60-100mg.  A typical espresso based Americano (1 shot + water to make 5oz) would also fall in that range.

http://www.aseanfood.info/Articles/11020406.pdf