I’ll bet you knew I was going to suggest Geocaching. There are many caches hidden inside buildings. Some are virtual caches where you have to follow clues… Even in the museums! Some clever, creative folks put them out for your enjoyment. And it’s free!
Metropolitan Museum of Art, just give them a dollar or two. I think Natural History is also a suggested price, but the planetarium would be a required fee. The small transit museum annex at Grand Central. Actually I don’t think the main transit museum is very expensive. Staten Island ferry if you like a boat ride. Just take it round trip if the rain isn’t bad. What about the Jaques Torres factory tour? Does that still exist? I think it was in Brooklyn, and then some yummy chocolates.
The wonderful public library on 5th avenue. Wonderful exhibits. I go there every time I’m in town. There’s always something new. And it’s absolutely free.
The little gems like the Frick (70th and Fifth), the Morgan Library (36th off Fifth Ave), and The Asian Society Museums (70th and Park) are wonderful and not so far apart.
The Frick alone will shore you up for years to come. And if you then want, you can then roller skate down the ramp at the nearby Guggenheim.