Can I disable Notify Activity Assignees for some Activities and/or user levels and leave it on for other Activities/user levels?
The extension mentioned by Peter in the answer is now moved into core and can be used from 4.7.30 version. See https://issues.civicrm.org/jira/browse/CRM-21380

It introduces a new setting which blocks some activity types to send an email to the assignees. 

To enable this, navigate to Administer -&gt; Customize Data and Screens -&gt; Display Preferences and select Notify Activity Assignees checkbox. It will then display a multi-select box to choose some activity type which you don't wish to send notifications for.