We propose a model for growing networks based on a finite memory of the nodes. The model shows stylized features of real-world networks: power-law distribution of degree, linear preferential attachment of new links, and a negative correlation between the age of a node and its link attachment rate. Notably, the degree distribution is conserved even though only the most recently grown part of the network is considered. As the network grows, the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors, coauthorship in science, and word synonyms. These highly clustered scale-free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks.