And each time you check in you get a new set of status benefits, such as two bottles of water, cookies etc (depending on the property). So for a three night trip, you can collect 3 stays (going A-B-A as suggested) as well as 6 bottles of water or whatever privilige is offered according to status).
also each time you check in (stay) you receive 500 FF miles thanks to the double dip - so 5 days of 'stays' vs 'nights' equals 2500 vs 500 miles !