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The problem of understanding the structure of transcendental objects has fascinated mathematicians for well over a century. Combinatorics provides an intuitive framework to study power series. A combinatorial family is associated to a power series in R{t} via its enumerative generating function wherein the number of objects of size n is the coefficient of t^n. Twentieth century combinatorics and theoretical computer science provided characterizations of classes with rational and algebraic generating functions. Finding natural extensions of these correspondences has been a motivating goal of enumerative combinatorics for several decades. This talk will focus on two well studied classes of transcendental functions: the differentiably finite and differentially algebraic.