I have a compiled list of all the guests that have attended different
events
that we have held in the last couple of years. We are trying to figure out
the loyalty of our guests; identifying those that have attended 2 or more
events. The guest names are in cells A3:A988. How do I come up with a
formula
so that the output is the name of anyone who has attended more than one
event
(meaning their name appears more than once in column A) and the number of
times they've attended an event, which is updated as more events/names are
added to the list?

This is a little different than what you asked for, but perhaps you will
find it useful. The way it works is you specify a name, and every occurrence
of that name will be flagged along with the number of total occurrences.
While I realize you probably have other columns filled in, this solution
assumes that column B will show the flagged rows and that cell C1 will be
where you specify the name to be searched for. Place this formula...

=IF(A1=C$1,COUNTIF(A$1:A$2000,"="&C$1),"")

in B1 and then copy it down through row 2000 (you can use a different
number, but I figured this would give you some growth potential). If you
want to specify a different maximum number of rows, just make sure you
change the 2000 in the formula to the maximum number of rows you will copy
the formula down through. Now, place the name you want to search on in C1
and hit the Enter key... each occurrence of that name will be flagged in
column B and the number shown is the total number of times the name occurs.

Rick