You can Union the polygon layer to itself (that is only add it to the Union feature list). Then dissolve the resulting polygon stacks so that all overlaps become a single polygon with a count of the overlaps involved in the dissolve (this may require setting a merge field rule).
In order to dossolve the stacked polygons you'll need to give them an ID that all identically overlapped polygons will share. You can do this bycreating the X and Y coordinates for the Unioned polygons and then concatentating into a new field, XY_id, before dissolving on that field. If the DIssolve does not directly give you a count field you can creat your own by adding a field "count", setting all feature values to 1, then settign the file merge rul to "sum" when dissolving.
Then convert to raster using the count field as the raster value.
Does any of that make sense? I can try to explain it better if not.