What is the highest floor of any New York building that may be reached by the general public for free?
The best option I've been able to find is Bar SixtyFive, on the 65th floor of Rockefeller Center:
The highest outdoor terrace bar in New York City creates an elevated lounge experience 65 stories above the landmark Rockefeller Center. Bar SixtyFive at Rainbow Room offers playful bistro fare and curated cocktails in a polished vintage gem framed by awe-inspiring urban views.
Bar SixtyFive is open weeknights from 5:00 PM until close for walk-ins, reservations and private events upon request. The kitchen is open until 11:00 PM Monday to Thursday and until close on Friday.
Serving guests 21 and over. Proper attire requested.
The bar can definitely be reached for free, though the staff there may not look kindly upon you if you just want to hang out for a while without buying a $14 glass of rosé. It is also debatable whether the restrictions on age and attire disqualify this as "open to the general public".
EDIT: Note that before 2001, the answer would probably have been Windows on the World. There had apparently been some thought given to putting a restaurant at the top of the new One World Trade Center building, but these plans were cancelled in 2011.