Force Sans Potential Is it possible to have a force without an associated potential energy? I know when a related potential energy exists, the relationship is $$U = -\int{F(x) dx}$$, but I am curious as to whether the one can exist without the other.
 A: If you think of potential energy in the framework of classical mechanics and gravitation, then yes. However...
Presence of force implies energy exchange, whether it is in its explicit form (i.e. a stone falling from the cliff) or implicit (i.e. friction force from the previous answer). A state of philosophical understanding of force these days, as far as I get it, is that any force is mediated by agents: photons, gravitons, gluons etc. These particles carry some form of energy with them and "deliver" it from "sender" to the "recipient", so to say. If one thinks about it this way, there is always some kind of field (potential) present, that mediates interaction. At this point it is the question about definitions, not actual phenomena...
P.S.: in the case of friction, this field is electrostatic. 
A: Yes and no. 
Friction is a great example of a force that doesn't have a well defined potential energy field associated with it. You can pull some strings and a create a faux potential (see Rayleigh dissipation function) to derive the frictional force from, but it doesn't behave the way a real potential would.
