If an object has no possible direction to go, does it still have potential energy? If a ball is sitting in a highly concave bowl that is as close to the ground as it can possibly be, and there is no external force able to act on it, and it is not able to move in any direction whatsoever, does it still have potential energy?
 A: Potential energy is not absolute; in other words, you can't say what a object's energy is, only how it would change if it moved. So in your example all we can say is that if it was higher its energy would be higher, but it's meaningless to ask what its energy is. You could say that it's zero or you could say that it's a million joules, it's just a reference point.
A: Yes, unless it is at absolute zero temperature, which is impossible. The atoms comprising the ball have plenty of potential energy, 
As opposed to the mechanical P.E. you are trying to eliminate in your question,  the bonds between the atoms in the molecular structure do have chemical potential energy.
Depending on the temperature of the ball, (usually proportional to it) there are stresses within the ball that in effect store energy, 
In classical terms, the ball may seem still, but even on this level, the ball will move about slightly, due to internal stresses, although how much detection effort you want to put into this is another question.

Image source: Molecular Composition of Rubber 
This a picture of the molecules within rubber, to take one example of the material a ball might be  made of. Each molecular group can be imagined as being joined together by spring-like forces,  and where you have springs, you have potential energy.
A: Yes, it has PE but a very small amount. Let's assume that the bowl is on a table. The difference in height between the ball being in the bowl on the table and on the table itself is the thickness of the bowl. 
A: In our world the situation isn't possible - you can't insulate against gravity so there is always gravity acting so the ball has gravitational potential energy.
However, in some other world (not our own), where the situation you describe holds, the ball wouldn't have potential energy; potential energy describes the possibility of motion ie into kinetic energy; in a world where the ball in the bowl cannot possibly move, not even theoretically, but is a wholly fixed element of the world it can't have potential energy; or rather to ascribe potential energy is not meaningful.
