This option is available on ArcGIS online, but I have not figured out if this can be acheived on the ArcGIS server.
Slippy map is a very easy way for developers to consume a map service and is widely supported. Your map service layer API on ArcGIS online look something like this:
https://tiles.arcgis.com/awsødkøaslkd/arcgis/rest/services/<service name>/MapServer/tile/{z}/{y}/{x}
Is this possible on ArcGIS Server?
https://wiki.openstreetmap.org/wiki/Slippy_map