  In the absence of spin-orbit coupling, the conventional dogma of Anderson
localization asserts that all states localize in two dimensions, with a glaring
exception: the quantum Hall plateau transition (QHPT). In that case, the
localization length diverges and interference-induced quantum-critical spatial
fluctuations appear at all length scales. Normally QHPT states occur only at
isolated energies; accessing them therefore requires fine-tuning of the
electron density or magnetic field. In this paper we show that QHPT states can
be realized throughout an energy continuum, i.e. as an "energy stack" of
critical states wherein each state in the stack exhibits QHPT phenomenology.
The stacking occurs without fine-tuning at the surface of a class AIII
topological phase, where it is protected by U(1) and (anomalous) chiral or
time-reversal symmetries. Spectrum-wide criticality is diagnosed by comparing
numerics to universal results for the longitudinal Landauer conductance and
wave function multifractality at the QHPT. Results are obtained from an
effective 2D surface field theory and from a bulk 3D lattice model. We
demonstrate that the stacking of quantum-critical QHPT states is a robust
phenomenon that occurs for AIII topological phases with both odd and even
winding numbers. The latter conclusion may have important implications for the
still poorly-understood logarithmic conformal field theory believed to describe
the QHPT.
