We consider properties of Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. This is a slight generalization of the situation first considered by Fornaess and Sibony in Random iterations of rational functions. Ergod. Th. & Dynam. Sys.11 (1991), 687–708. The classical definition of hyperbolicity has a natural extension to our setting, and we show that many of the classical results concerning hyperbolic Julia sets can indeed be generalized. We introduce two important ideas in order to prove these results—the use of Cantor diagonalization, and how the Julia set moves in the Hausdorff topology as the sequence generating it changes.