Quantifying the complexity of large graphs requires measures that extend beyond predefined structural features and scale efficiently with graph size. This work adopts a generative perspective, modeling large networks as exchangeable graphs to quantify the information content of their generating mechanisms via graphon entropy. As a graph property, graphon entropy is invariant under isomorphisms, making it an effective measure of complexity; however, it is not directly computable. To address this, we introduce a suite of graphon entropy estimators, including a nonparametric estimator for broad applicability and specialized versions for structured graphons arising from well-studied random graph models such as Erdős-Rényi, Chung-Lu, and stochastic block models. We establish their large-sample properties, deriving convergence rates and Central Limit Theorems. Simulations illustrate how the nonparametric graphon entropy estimator captures structural variations in graphs, while real-world applications demonstrate its role in characterizing evolving network dynamics.