You're just moving the clone instead of duplicating (see the comments left below your question).
$("#add-student-link").click(function() {
++student_number;
$("#student-1").clone().attr('id', 'student-' + (student_number)).addClass("additional-student").appendTo('form.reg');
});
If you're cloning to keep a copy that's clean (could be input fields in the element for all I know) then clone the clone.
var orig_student = $("#student-1").clone().attr('id','');
$("#add-student-link").click(function() {
++student_number;
orig_student.clone().attr('id', 'student-' + (student_number)).addClass("additional-student").appendTo('form.reg');
});