At least once you should take a nap during the day, and go out at night and plan to spend all night checking out the nightlife.
I used to live in Fontainebleau - its a nice day trip to the countryside and there are some nice buildings and gardens to see.
Last time I was there was '83 and I spent a week sleeping on the streets - I had about ten bucks to make it through the week. Long story.
The big thing about Paris is just walk around a lot - there are so many cool things to see off the beaten path. Do the main stuff, take the pics then throw the map away, rent a bike and head off somewhere. Make sure you spend some wandering time on the left bank - thats where I remember the bohemians and artists hanging out. Probably lots of cool art galleries there.
You won't be bored.
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