Hi Royce,
I tested real quick with this service: http://mesonet.agron.iastate.edu/cgi-bin/wms/goes/east_ir.cgi?
This map can be added as an operational layer to an ArcGIS.com web map and projects properly. The best way to do this would be to add these layers client side.
Performance is a big reason for preventing 'cascading' map services, but there are also questions of service owership and attribution to consider.
I'm unsure the context, but I can say that at least with Esri-powered WMS from ArcGIS for Server, at 10.3 any cached map service also has the WMTS capability enabled. In this case, since its coming from the MapServer product, I'd definitely just add it client side using an app built with the JS API.
At 10.0, this could be worked around by publishing a 'Classic' map service instead of an 'Optimized' map service. At 10.1+, all services are optimized.