Yesterday we took the train out to the town and Palace of Fontainebleau. Because it's an Isle-de-France train Daddy's Zone 5 Navigo entitled him to the trip so we just had to buy my ticket. The man at the guichet sold me a zone 5 daypass, which is €2 cheaper than the aller-retour, so that's a good tip.
This trip is clearly what is done by locals with children on the weekends, it was lovely for us and with the exception of some tourists everyone else who was there was with kid(s). We had full run of the grounds and walked and played and saw some enormous carp and fed some swans and had a lovely afternoon.
We went into the town for a hot chocolate and discovered a stunning "chestnut" chocolate (must post photo here). Walking through the town it reminded us of Wellesley, MA. We plan to go back and picnic at the chateau when the leaves are out and we can ride bikes.