An algebra of bounded linear operators on a Hilbert space is called strongly compact whenever each of its bounded subsets is relatively compact in the strong operator topology. The concept is most commonly studied for two algebras associated with a single operator T: the algebra alg(T) generated by the operator, and the operator's commutant com(T). This paper focuses on the strong compactness of these two algebras when T is a composition operator induced on the Hardy space H² by a linear fractional self-map of the unit disc. In this setting, strong compactness is completely characterized for alg(T), and "almost'' characterized for com(T), thus extending an investigation begun by Fernández-Valles and Lacruz [A spectral condition for strong compactness, J. Adv. Res. Pure Math. 3 (4) 2011, 50-60]. Along the way it becomes necessary to consider strong compactness for algebras associated with multipliers, adjoint composition operators, and even the Cesàro operator.