We study a network formation game where n players, identified with the nodes of a directed graph to be formed, choose where to wire their outgoing links in order to maximize their PageRank centrality. Specifically, the action of every player i consists in the wiring of a predetermined number di of directed out-links, and her utility is her own PageRank centrality in the network resulting from the actions of all players. We show that this is a potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node i to link to other nodes that are at incoming distance more than di from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the homogeneous case, where players all have the same number d of out-links, we characterize the structure of the potential-maximizing equilibria, and in the special cases d = 1 and d = 2, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks. Funding: This research was carried out within the framework of the Ministero dell’Università e della Ricerca (MIUR)-funded Progetto di Eccellenza of the Dipartimento di Scienze Matematiche G. L. Lagrange, Politecnico di Torino [CUP: E11G18000350001]. It received partial support from the MIUR-funded project PRIN 2017 “Advanced Network Control of Future Smart Grids” and from the Compagnia di San Paolo.