All beach chairs at Atlantis Beach (aka Cabbage Beach) and the Cove Beach are complementary, but they make you sign in with a orange shirted staff person. I noticed some people avoided getting beach chairs because thought there was a charge like at some hotels. They let you use the Paradise Beach in front of the Cove and Reef, but they don't allow you to use their beach chairs. You have to drag one over from the Cove Beach and there are limits where they will let you put it.
All watersports (i.e. paddle boats, kayaks, all motorized stuff) and water equipment (sand toys, snorkels) costs extra except life jackets for small children. Those are in big brown trunks near pools.
All Aquariums are free, unless they are in a restaurant or club with restricted access. The Dig, the big themed Aquarium exhibit is free, though sometimes there is a guard to check keys or wristbands. You are allowed to visit a fenced patio/viewing area at Dolphin Cay, but the views are somewhat distant from the Dolphins. There are no underground viewing areas or catwalks to get a close view if you aren't in the water with them.
There are daily tours of the aquariums, but I never did them. I got conflicting information on whether they were free for guests or if they cost $25 or so.
The gym at Harborside is free, but the Spa gym near the Royal towers has a fee (~$25 per day) and gives you access to common areas of the Spa including the salt water pool.
All kid activities like Club Rush, remote control cars, pottery, and the Kids club cost extra. Though the movies are free, concessions aren't. Rated G in mornings, PG in afternoons, PG13 late afternoon/early evening, R late at night. The comedy club was not free and very expensive.
There is a hair braiding and craft huts near the beaches too, which aren't free. Unfortunately, they don't stay in those areas and harass you on Atlantis Beach, but not the Cove and Paradise Beach.
The Library at the Coral Tower is free and Harborside had a take a book leave a book area too.
There is supposedly a free bus that takes people from the Casino to other hotels on PI and Nassau, but hotel guests aren't supposed to use it.
One amenity that I would have liked but never saw was water fountains. That would have been nice and saved me a lot of money.