  The degree excess function $\epsilon(I;n)$ is the difference between the
maximal generating degree $d(I^n)$ of a homogeneous ideal $I$ of a polynomial
ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically
linear function $d(I^n)$. It is shown that any non-increasing numerical
function can be realized as a degree excess function, and there is a monomial
ideal $I$ whose $\epsilon(I;n)$ has exactly a given number of local maxima. In
the case of monomial ideals, an upper bound on $\epsilon(I;n)$ is provided. As
an application it is shown that in the worst case, the so-called stability
index of the Castelnuovo-Mumford regularity of a monomial ideal $I$ must be at
least an exponential function of the number of variables.
