If you are feeling still more adventurous, rent a car on day seven and spend your last day in the area driving Germany's legendary Romantische Strasse (Romantic Road). Stretching 180 miles, it is full of old towns and villages that make you ponder what Munich might have been back in the day.
One day will only allow you to drive a small portion of the road, but you will still catch the spirit of the experience. You can easily forgo a day or two in Munich and expand your trip to allow more time on the Romantic Road.
If you do so, make Neuschwanstein one of your stops. It is home to another of Mad King Ludwig II's famous castles, though he was removed from the throne (due to his alleged insanity) before it was completed. The tour includes the king's apartments, the unfinished throne room, the king's bedroom, and the highlight of the tour, the 4th floor Singer's Hall.
After Ludwig's mysterious death in 1886 (technically by drowning, but conspiracy theories abounded), the castle was opened to the public for tour. Interestingly, the castle is considered the inspiration for many structures seen in pop culture, including Sleeping Beauty's Castle at Disneyland in California.