Is there a max terminal velocity much less than speed of light? If you had an object falling into, say, a neutron star which had a very great gravity, is there some reason other than an atmosphere that would prevent the object from reaching any arbitrary speed up to that of light or is it pretty simple: if the star was massive enough, a bowling ball or whatever would be moving at half or 3 quarters of light speed when it hit the surface?
What I am thinking of is that the inertia of the ball might grow so fast that it reaches equilibrium or acceleration at least can no longer be calculated using non-relativistic formula.
 A: The maximum velocity of an object falling from rest down to a neutron star (or planet, or anything really) is simply the escape velocity from the surface of that star or planet. There's no other limit. In a vacuum it will keep accelerating until it hits the surface.
As to whether the acceleration has to be calculated using relativity: to get the exact answer you always have to use the relativistic formula, but as long as the velocity $v$ is significantly less than $c$ the traditional (non-relativistic) formula will give a good approximation, with the error approaching 0 as $v$ approaches 0.
A: For an object falling from a long way from the neutron star, the speed measured at the surface will be $c\sqrt{r_s/r_n}$, where $r_s = 2GM/c^2$ is the Schwarzschild radius for an object of mass $M$ and $r_n$ is the radius (or more technically, the radial coordinate) of the neutron star surface.
This formula is derived directly from the Schwarzschild metric for a non-spinning, spherically symmetric mass, so is relativistically correct. Note that it agrees with the Newtonian escape speed of $\sqrt{2GM/r_n}$.
Since for all neutron stars, $r_n > r_s$, (by about a factor of 2, otherwise they would be black holes), then the impact velocity must always be less than $c$ the speed of light, but could certainly exceed $c/2$.
If the object falls through a vacuum, there will not be anything to prevent it reaching this speed. Neutron star atmospheres are of order 1 cm thick, compared with $r_n \sim 10$ km, so this offers no appreciable braking before impact. Of course if the body had a net charge, then it might be affected by the strong electromagnetic fields outside the neutron star.
