  We study the infinite-temperature properties of an infinite sequence of
random quantum spin chains using a real-space renormalization group approach,
and demonstrate that they exhibit non-ergodic behavior at strong disorder. The
analysis is conveniently implemented in terms of SU(2)$_k$ anyon chains that
include the Ising and Potts chains as notable examples. Highly excited
eigenstates of these systems exhibit properties usually associated with quantum
critical ground states, leading us to dub them "quantum critical glasses". We
argue that random-bond Heisenberg chains self-thermalize and that the
excited-state entanglement crosses over from volume-law to logarithmic scaling
at a length scale that diverges in the Heisenberg limit $k\rightarrow\infty$.
The excited state fixed points are generically distinct from their ground state
counterparts, and represent novel non-equilibrium critical phases of matter.
