I'm surprised there's no mention of the Berkeley, CA Adventure Playground (http://www.ci.berkeley.ca.us/adventureplayground/), in continuous operation since 1979.

I have fond memories of being there as a young child. Want a hammer? Pick up 10 stray nails and trade them for a hammer. Then go smash things, attach things, whatever you like.

There's a zip line and a giant cargo net to climb. It's an incredibly empowering experience for children.