Can I get a new random number per ray in the node editor? In the node editor, I am looking for a way to get a random number (in [0,1)) per ray.  The object info node's random output just gives a random number per object, the number never changes.  I am looking for a way to get a new number every time the shader is sampled.

I am aware of this question, however that is not quite what I am looking for.
There's enough randomness in the Position from the Geometry node. It will almost never be exactly the same for two samples, but you have to apply a hash function to convert it to a random number in the [0,1) range.

The simplest solution would be to use a Voronoi or Noise Texture node with a very large Scale.

Another solution is to make your own hash function using math nodes, replicating for example this GLSL code.