I'm having the exact same problem on several computers, and we're trying to publish to ArcGIS Online, not our own server, so licensing can't be the problem. Plus, the problem is that Mapping (i.e., dynamic map service) is not showing up. That is the core capability, not an advanced one that would require an extra license.
We're using 10.2.2.
If you're publishing a hosted service from ArcGIS Desktop to ArcGIS.com, there are only two options: Feature Services and Tiled Services.
About hosted web layers—ArcGIS Online Help | ArcGIS : types of hosted web layers.