SQL query to get the user, counting the minimum number of records
Add a WHERE clause to filter the rows where last_edit_user is null. Then put another query around it, to get the minimum count. Use this as a subquery in the HAVING clause to get only the ones with the count equal to the minimum.

SELECT last_edit_user,
       count(last_edit_user)
       FROM edited_elements_table
       WHERE last_edit_user IS NOT NULL
       GROUP BY last_edit_user
       HAVING count(last_edit_user) = (SELECT min(c)
                                              FROM (SELECT count(last_edit_user) c
                                                           FROM edited_elements_table
                                                           WHERE last_edit_user IS NOT NULL
                                                           GROUP BY last_edit_user) x);