Is it possible to stack perfect spheres? Is it possible to stack a perfect sphere on top of another? It is easy to stack a cube on top another, but as the faces of the shapes increase, it seems more and more difficult to stack. So, is a sphere with (infinite sides or no sides?) "stackable?" This scenario does not have to be real-world based, but rather in a "perfect" environment.

A sphere technically has infinite sides if it is "perfect", because it is entirely smooth.  This doesn't ever occur in reality, as once we magnify any given edge of matter that is smooth we will find imperfections in the material.  Even if we didn't, then at the atomic level we would find the nuclear structure that makes up the matter isn't perfectly smooth, nor is the magnetic field around the atom devoid of fluctuation.  
So let's say we have two perfect spheres made of unbelievium.  These exist in a perfect vaccuum, with absolutely no other forces acting on them.  If we are making the binary assumption of stacked versus not stacked, we have to assume they are both at the same time (Quantum Mechanics).  Observation would require an influencing force.  
So moving past that flaw in the attempt, we'll say that whatever method by which we observe (light, etc) passes through our unbelievium harmlessly and without any influence.  Now we're pinned in behind the definition of a stack.  Without a reference point for "down", you can't define a stack.  
So to get past that, lets say the observer's sense of being upright is "up and down".  Since we're in a vacuum, stacking is possible; reducing the balls to a transverse velocity of 0 while touching each other would constitute a stack. 
I break apart the question above because of the nature of the question.  Technically, not only is the answer not physically possible in our reality, but the question effectively contradicts itself when attempting to answer it.
