I want to show the number of days between the current date (today)
and the hiring date in order to provide an indication of how many days
is the anniversary away. Any suggestions?

Here is my offering (assumes "hire date" is in A1)...

=DATEDIF(TODAY(),DATE(YEAR(TODAY())+(DATE(YEAR(TOD AY()),MONTH(A1),DAY(A1))<TODAY()),MONTH(A1),DAY(A1 )),"d")

While character count is not an absolute metric, I note that this formula
is the shortest one offered so far except for Biff's, mine is one character
longer than Biff's formula; but, as Sandy pointed out, Biff will have to
modify his formula to account for the problem Sandy has pointed out, so we
will see how things shake out on the character count later.

Rick