CurrentUser function in Reports?
It's not quite what you're looking for but if you install the Extended Reports extension then you can do:


Go to Administer - CiviReport - Create Report from Template.
Pick extended report - activities.
Pick some columns for the activity.
On the Display Options tab, check the box to display as a tab on the contact record, and for the box below that choose assignee.
Click refresh results.
From the actions dropdown choose Create Report.


Then on any given contact they'll have a tab that shows their assigned activities.