As the number of throws goes to infinity, the velocity as seen by the point you call "fixed" approaches c, the speed of light.
It's not hard to solve the continuous form of this problem, with lots of throws (N) each adding just a little v, where v is a small fraction of c. Then after N throws the velocity of the ball in the "fixed" reference frame will be c*tanh(Nv/c), where "tanh" is the hyperbolic tangent function.
Here's a picture of how the velocity will change as a function of Nv/c. 
Mike W.
(published on 06/19/2013)
