<p>I am going to design and build a 3D printer. I want the highest quality and accuracy so nothing except that is important for me. Which cartesian design has the highest quality and accuracy? CoreXY, Prusa, or Gantry (Ultimaker)?</p>
<p>Also, is it better to have a nozzle that moves in just direction &quot;X&quot;, directions &quot;X and Y&quot;, or &quot;X, Y and Z&quot;?</p>

The tradeoffs in these systems are all about quality achievable at particular speed and acceleration profiles. If you really don't care about speed at all and want maximum accuracy, you probably want some type of Cartesian setup with no belts, only rigid lead screws which you can take to as fine a pitch as you like, and you can make all the parts as rigid as you like because mass doesn't matter (since acceleration doesn't).
Note however that extrusion accuracy is the limiting factor to quality and dimensional accuracy in even a half-decent printer. Rather than trying to design something with "perfect" spatial kinematics for quality from the outset, I think you should look at existing printers, figure out what about them isn't meeting your quality needs, and start from there to improve. You should also figure out what your speed constraints will be, even if they're only minimal.