We consider twist maps on the plane (like the ping-pong map) with non-periodic angles, where typically bounded and unbounded motions co-exist. For the latter case we prove a theorem which shows that in the analytic setting the growth rate is at most logarithmic, and furthermore an example of a system is given where all orbits grow at this rate. Moreover, we determine the optimal growth rate for a ping-pong with finite regularity. (This is joint work with R. Ortega.)