Why do tokamaks have so few toroidal field magnets? From the ITER webpage itself, the tokamak has only eighteen toroidal field magnets. If the magnetic field is directly and linearly dependant on the number of coils around the torus, then why did the best plasma physicists choose to have so few? Wouldn't many problems, such as the maximum current density of the coils, or the systems necessary to keep these coils cooled be solved by having many more, thinner coils? What qualities made fewer superconducting coils more attractive than many?
 A: Intensive optimization routines go into planning the number of toroidal field magnets. Technically, groups of thin coils like the ones you propose are actually what tokamaks use, but you may be thinking from a purely physical (i.e. not practical) perspective. 
Groups of ideally thin conducting coils grow considerably in size by the need for (often central) cooling channels and insulating tapes and covers. The size of every coil unit is further increased by tensile stress considerations which demand thick stainless steel cases and robust support structures. If you take a look at resulting toroidal field coils in their cases, the thickness of each unit is easily above 1 meter and significantly more when support structures are included for attachment in the tokamak vessel. 
Consider a very large reactor like ITER has a mid-plane circumference below 40 meters. You can imagine a spatial limit in the number of such coil cases that you can stack toroidally side by side toward the center of the tokamak. 
From early optimization calculations for ITER focused purely on the magnetic field, 24 toroidal field coils were required. I assume that would have made use of the entire space to stack coil cases. In practice, the number of coils has to be further reduced for access considerations: the space has to be shared with a multiplicity of large heating systems (NBI, ECRH, ICRF, etc.), fueling systems and even more diagnostic systems. So the number of coils was subsequently reduced for ITER to 20 and, finally, to 18.
All in all, I wouldn't say plasma physicists oppose the idea of a large number of thin conducting coils to increase the toroidal field in tokamaks. I would say they embrace the method, but real world considerations get in the way.
