On any compact Riemannian manifold $(M, g)$ of dimension $n$, the
$L\sp 2$-normalized eigenfunctions $\{\phi\sb \lambda\}$ satisfy
$\lvert \rvert\phi\sb \lambda\lvert \rvert\sb\infty\leq C\lambda\sp
{(n-1)/2}$, where $-\Delta\phi\sb \lambda=\lambda\sp 2\phi\sb
\lambda$. The bound is sharp in the class of all $(M, g)$ since it is
obtained by zonal spherical harmonics on the standard $n$-sphere $S\sp
n$. But, of course, it is not sharp for many Riemannian manifolds, for
example, for flat tori $\mathbb {R}\sp n/\Gamma$. We say that $S\sp
n$, but not $\mathbb {R}\sp n/\Gamma$, is a Riemannian manifold with
maximal eigenfunction growth. The problem that 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
$\mathscr {L}\sb x$ of geodesic loops at $x$ has positive measure in
$S\sp \ast\sb xM$. We show that if $(M, g)$ is real analytic, this
puts topological restrictions on $M$; for example, only $M=S\sp 2$ or
$M=\mathbb {R}P\sp 2$ (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
$\mathscr {L}\sb x$ has positive measure for an open set of $x$ but
which does not have maximal eigenfunction growth; thus, it disproves a
naive converse to the main result.