  It is well known that the low energy expansion of tree-level superstring
scattering amplitudes satisfies a suitably defined version of maximum
transcendentality. In this paper it is argued that there is a natural extension
of this definition that applies to the genus-one four-graviton Type II
superstring amplitude to all orders in the low-energy expansion. To obtain this
result, the integral over the genus-one moduli space is partitioned into a
region ${\cal M}_R$ surrounding the cusp and its complement ${\cal M}_L$, and
an exact expression is obtained for the contribution to the amplitude from
${\cal M}_R$. The low-energy expansion of the ${\cal M}_R$ contribution is
proven to be free of irreducible multiple zeta-values to all orders. The
contribution to the amplitude from ${\cal M}_L$ is computed in terms of modular
graph functions up to order $D^{12} {\cal R}^4$ in the low-energy expansion,
and general arguments are used beyond this order to conjecture the
transcendentality properties of the ${\cal M}_L$ contributions. Maximal
transcendentality of the full amplitude holds provided we assign a non-zero
weight to certain harmonic sums, an assumption which is familiar from
transcendentality assignments in quantum field theory amplitudes.
