I guess you've tried this method: https://gis.stackexchange.com/questions/51929/how-to-count-and-rasterize-polygon-overlaps-in-arcgis-desktop

16,400 polygons isn't that many. However, one potential solution is to simply do a regular Spatial Join. In the `ArcMap toolbox, > Analysis Tools -> Overlap -> Spatial Join`.

Set both the `target` and `join` features to the same dataset and specify an output. Leave the rest of the settings.

After a few moments you should get back a shapefile that contains a "join count" column. Subtract 1 from this (as obviously each feature should intersect itself), and that should be the number of "overlaps "(actually intersects) for each polygon.

I just performed it on