I just returned from the BOE 14 day tour last week. You're going to love it. Here is some of what we did on our free time -- none of which was pre-planned.
Paris - Musee d'Orsay (highly recommended) and some other sites, like inside of Notre Dame, Place de la Concorde, climbing the Arch de Triomphe at sunset. We also did the Eiffel Tower the morning before our tour began (2 PM that day). Not sure you could meaningfully do Versailles during the free afternoon.
Beaune - Dinner (highly recommend Le P'tit Paradis if you can get a reservation), Hospices de Beune, walking the Ramparts.
Lauterbrunnen -- Our free day was crystal clear, so we followed our guide up Schilthorn and Kleine Scheidegg for spectacular views (do NOT opt out of this) and the James Bond museum (just kidding -- there really is one at Schilthorn, but totally skippable). Trummelbach Falls was only so-so after the earlier spectacular views.
Munich -- We did Dachau, the only ones in our group to do so. Highly recommended, but not a feel-good sort of activity.
Venice -- We did the Doges Palace (including Bridge of Sighs), Correr Museum, and ate too much gelato (hold out for the good stuff).
Florence -- Climbed a hill across the river for spectacular views of Brunelleschi's Dome.
Rome -- We stayed two extra days, so took our time but ultimately covered the Scavi Tour under the Vatican (highly recommended, but need to reserve months in advance), Capitoline Museum, Spanish Steps, walked all over the place, and, again, ate too much gelato.
Hope this helps. Enjoy your tour, but there's very little risk that you won't.