Does the potential energy of a single object exist?

I know that potential energy exists for a system of objects. But why? Is there no potential energy for a single object system? I assume that there is.

Consider the earth-object system. Suppose an external force lifts the object above the earth. Then the work done by it gets stored in the earth-object system as potential energy.Right?

Now, my question is that is the potential energy always zero for a single object system.

MY REASONING

If we consider the object only as the system and lift it above the Earth, we are doing positive work on it and filling it with energy and the gravitational field of the Earth is doing negative work on it and draining it of energy. Thus whenever the object is moving it has kinetic energy which is fine. But when it is not moving,i.e. when its weight is equal to the force applied by us, the net work done is zero and so the object should not have any kinetic energy. Does it have potential energy?