Hi Nick,
We've done something similar, but without using rasters.
First, add an integer item called COUNT to your polygon feature.
Then use the Intersect tool to intersect your polygon layer on itself.
Then use the Dissolve tool, and add the COUNT item as the Statistics field and set the Statistics Type to SUM.
This should output a polygon layer with the number of overlapping features captured in the COUNT field.
I hope this helps,
Greg Wise
Department of Planning and Local Government
South Australia