A key test of whether an API is open is whether it is used to enable services that are not hosted by the API provider...
Kevin Marks points out that the definition is too weak, a point which Tim concedes.
Regardless, one aspect of REST that gets frequently overlooked is Code on Demand. From Roy's Thesis:
In the code-on-demand style, a client component has access to a set of resources, but not the know-how on how to process them. It sends a request to a remote server for the code representing that know-how, receives that code, and executes it locally.
It's almost as if the web were designed for this stuff.