The potential energy of an object has to be absolute, a constant in some sense. The (gravitational) potential energy is a function of position so it varies asobjects move around inspace.
When dealing with gravity, the rate at which kinetic energy changes into potential energy will decrease with distance (height). If you throw something far enough that it goes beyond the earth and never comes back (like in your original question), the rate at which kinetic energy changes to potential energy will continue to slow down, but never reach zero. So the trick to getting something to escape earth's gravity is to make sure it is moving fast enough to never reach that zero point discussed above. So the object will always have some kinetic energy and some potential energy.