  We consider word complexity and topological entropy for random substitution
subshifts. In contrast to previous work, we do not assume that the underlying
random substitution is compatible. We show that the subshift of a primitive
random substitution has zero topological entropy if and only if it can be
obtained as the subshift of a deterministic substitution, answering in the
affirmative an open question of Rust and Spindeler. For constant length
primitive random substitutions, we develop a systematic approach to calculating
the topological entropy of the associated subshift. Further, we prove lower and
upper bounds that hold even without primitivity. For subshifts of non-primitive
random substitutions, we show that the complexity function can exhibit features
not possible in the deterministic or primitive random setting, such as
intermediate growth, and provide a partial classification of the permissible
complexity functions for subshifts of constant length random substitutions.
