Finding 2nd order polygon neighbors (and 3rd, 4th etc.) using ArcGIS for Desktop?

I have a shapefile of contiguous polygons and for each pair of polygons, I need to know how many polygons are between them, following an neighboring/bordering approach. The output is likely to be a non-spatial table.

For example, for this group of polygons:

The output would be something like this:

and so on...

I can use ArcGIS for Desktop for that.

My idea of solution so far:

A model where, for each polygon, I run Polygon Neighbors (Analysis), merge with all the neighbors and run it again, until all there's only one polygon. However, I have little knowledge on coding/scripting, so I'm unsure on how to build this model exactly and if there's a simpler way.