Fontainebleau sits in a part of the Île-de-France that is dense with notable destinations, all reachable within an hour by car.
Vaux-le-Vicomte (20 km north) is a château that predates Versailles and, many argue, surpasses it in elegance. Built for Nicolas Fouquet, Louis XIV's finance minister, it was so magnificent that Louis had Fouquet arrested for presumption and hired his entire design team to build Versailles. The gardens, laid out by Le Nôtre, are among the finest in France. Visiting by candlelight on Saturday evenings in summer is exceptional.
Moret-sur-Loing (11 km east) is a well-preserved medieval town on the Loing river, with fortified gates, a Romanesque church, and a riverfront that the painter Alfred Sisley made his subject for the last twenty years of his life. It is pretty and uncrowded, and a good hour's wander.
Provins (45 km northeast) is a UNESCO-listed medieval market town, one of the great towns of the Champagne fairs, with intact ramparts, a twelfth-century keep, and underground tunnels. It retains far more of its medieval fabric than most French towns and is significantly less visited than it deserves.
Versailles (60 km northwest) is an obvious pairing for those doing a circuit of the great royal residences, though the contrast is instructive: Versailles is overwhelmingly grand and anonymous; Fontainebleau is layered, intimate, and historically richer.