  We give a systematic expansion of the crypticity--a recently introduced
measure of the inaccessibility of a stationary process's internal state
information. This leads to a hierarchy of k-cryptic processes and allows us to
identify finite-state processes that have infinite crypticity--the internal
state information is present across arbitrarily long, observed sequences. The
crypticity expansion is exact in both the finite- and infinite-order cases. It
turns out that k-crypticity is complementary to the Markovian finite-order
property that describes state information in processes. One application of
these results is an efficient expansion of the excess entropy--the mutual
information between a process's infinite past and infinite future--that is
finite and exact for finite-order cryptic processes.
