The Washington Redskins. I dont know how many but a lot. Edit: It has been 26 coaches since 1932 I believe. Should be the most but not sure.
As of May 2014, the Pittsburgh Steelers has had a total of 17 head coaches. The current coach has been with the team for 16 years.
that would be Alabama
1 head coach (hence the name "head") but can have up to as many 'other' coaches. (abviousley not anything above 10)
The number of assistant coaches varies from team to team but here is what I understand the standard most teams carry. There are two assistant coaches on the bench along with the head coach. One assistant is for offense and the other for defense. There is also a goaltending coach who watches the game from a pressbox, "bird's eye" view and sometimes does doubles as the video coach. There is also a strength and conditioning coach. I have heard of teams carrying additional coaches for developing propects and are assigned to the minors and junior teams.
There are 32 head coaches (one for every team) and a lot of assistants.
As of the start of the 2007 season, there is no limit to the number of coaches a team may have.
The London Wasps rugby team have had various head coaches appointed over the years since the club was established in 1867. Some of the coaches included Rob Smith, Warren Gatland, Tony Hanks, Leon Holden and most currently Dai Young.