  The generic limit set of a dynamical system is the smallest set that attracts
most of the space in a topological sense: it is the smallest closed set with a
comeager basin of attraction. Introduced by Milnor, it has been studied in the
context of one-dimensional cellular automata by Djenaoui and Guillon,
Delacourt, and T\"orm\"a. In this article we present complexity bounds on
realizations of generic limit sets of cellular automata with prescribed
properties. We show that generic limit sets have a $\Pi^0_2$ language if they
are inclusion-minimal, a $\Sigma^0_1$ language if the cellular automaton has
equicontinuous points, and that these bounds are tight. We also prove that many
chain mixing $\Pi^0_2$ subshifts and all chain mixing $\Delta^0_2$ subshifts
are realizable as generic limit sets. As a corollary, we characterize the
minimal subshifts that occur as generic limit sets.
