  Given a knot $K$ in a closed orientable manifold $M$ we define the growth
rate of the tunnel number of $K$ to be $gr_t(K) = \limsup_{n \to \infty}
\frac{t(nK) - n t(K)}{n-1}$. As our main result we prove that the Heegaard
genus of $M$ is strictly less than the Heegaard genus of the knot exterior if
and only if the growth rate is less than 1. In particular this shows that a
non-trivial knot in $S^3$ is never asymptotically super additive. The main
result gives conditions that imply falsehood of Morimoto's Conjecture.
