Functional iteration, the process of forming a sequence x₀, x₁ = f(x₀), x₂ = f(x₁) = f(f(x₀)) = f²(x₀), . . ., by repeated application of a function f, is fundamental to the approximation of solutions to nonlinear equations and plays an important role in the study of nonlinear dynamics. In this note we examine another aspect of functional iteration: the global behavior of the sequence f, f², f³, . . ., fⁿ, . . . of composite functions as n → ∞. This topic can be presented to students at many levels using computer graphics to stimulate conjectures that can then be investigated theoretically.