  On any compact Riemannian manifold $(M, g)$ of dimension $n$, the
$L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy
$||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta
\phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of
all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard
$n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds,
e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a
Riemannian manifold with maximal eigenfunction growth. The problem which
motivates this paper is to determine the $(M, g)$ with maximal eigenfunction
growth. Our main result is that such an $(M, g)$ must have a point $x$ where
the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in
$S^*_x M$. We show that if $(M, g)$ is real analytic, this puts topological
restrictions on $M$, e.g. only $M = S^2$ (topologically) in dimension 2 can
possess a real analytic metric of maximal eigenfunction growth. We further show
that generic metrics on any $M$ fail to have maximal eigenfunction growth. In
addition, we construct an example of $(M, g)$ for which ${\mathcal L}_x$ has
positive measure for an open set of $x$ but which does not have maximal
eigenfunction growth, thus disproving a naive converse to the main result.
