It is important to note that whenever one of these people ascends to the Presidency through the Succession Act, he immediately fills all vacant positions in this order with new appointees. So, in order for say, the Secretary of Labor to become president, all 10 people before him have to die all at once (or at least, before they can appoint new officers for the vacant positions). Repeated deaths of only the President would just involve the Vice President assuming office, and appointing a new Vice President, who would then assume office on his death, ad nauseam.