Show the number of the calendar day
Include an option to show the number of the Gregorian calendar day of the year on the day. View of this number should be in different corner of the day control/box, and in a different/lighter shade than the day of the month.
Might not be what you want exactly, but I mitigated this need by adding a calendar specfically for this purpose:
Chris H commented
Like the week number, this information is useful in a naming convention for uniqueness when naming things that must be unique and generic at the same time, like webpage urls.