> Could you expand on why you think it approaches a constant, presumably a non-zero constant

Because the gravitational potential between two massive objects is equal to the amount of energy it would take to separate them (from touching) out to their current distances, and equal to the amount of kinetic energy they'd have when they impacted if they started from rest and eventually came together. This is not zero for any two massive objects that are any distance apart.

If you pulled two objects further and further apart in R3, the force required at each point would drop off proportional to 1/r^2, and the total energy would be the integral of that (let's assume total mass == 1). This integral approaches a constant: for a total mass of 1 and a starting (or "touching") distance of 1 (you can't use 0 or you can't pull them apart), this integral is 1. So no matter how far apart you pull them, potential never exceeds 1. But it must be nonzero, because you could extract energy from releasing them and letting them come together.

In R2, the force required to pull them apart at each point drops off as 1/r, so the integral grows as Ln(r), and so has no upper bound.

I recognize we basically agree on all this, I'm just clarifying why I consider the limit of the potential to be nonzero as distance approaches infinity. You could extract energy from the system with a "water wheel" setup, at least at every finite distance. You could argue "but not at an infinite distance, because they'd never come together, their attraction force is 0, and where would you put the water wheel?" but that problem only comes up at infinity. We're talking about the limit as r approaches infinity, so this problem never actually comes up.