  A \emph{quasi-polynomial} is a function defined of the form $q(k) = c_d(k)
k^d + c_{d-1}(k) k^{d-1} + ... + c_0(k)$, where $c_0, c_1, ..., c_d$ are
periodic functions in $k \in \Z$. Prominent examples of quasi-polynomials
appear in Ehrhart's theory as integer-point counting functions for rational
polytopes, and McMullen gives upper bounds for the periods of the $c_j(k)$ for
Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be
sharp, but sometimes smaller periods exist. We prove that the second leading
coefficient of an Ehrhart quasi-polynomial always has maximal expected period
and present a general theorem that yields maximal periods for the coefficients
of certain quasi-polynomials. We present a construction for (Ehrhart)
quasi-polynomials that exhibit maximal period behavior and use it to answer a
question of Zaslavsky on convolutions of quasi-polynomials.
