I only can comment for small school where the majority of our coaches are stipend only. I am working hard to change that and you will soon see why. Here is the list off the top of my head:
Creation of and tracking of budget for each sport
Posting job openings/reviewing resumes/interviewing/hiring coaches. Takes much more time than I thought.
Supervising/evaluating/training or providing training to/replacing coaches.
Managing use of numerous facilities in and out of season such that the school maximizes incoming revenue from its resources and students maximize benefits from use of these resources. Attending all staff meetings during which you learn that drama will be using the stage in your gym and you will need to find another place for your middle school teams to practice, or they will be practicing all together, boys and girls on half a court. Communicate all such information to your coaches, read happy and supportive emails from your coaches, etc. Probably takes most time of any activity.
Managing all aspects of adequately equipping teams and facilities, including uniforms, equipment necessary for play (these first two difficult for growing school), sport-specific training equipment, general strength and conditioning equipment, playing field prep equipment, and maintenance and inventory of the preceding.
School registration for all sports. Includes letter and forms going out, receipt and filing of all necessary forms, money to accounting, invoices to those who have not paid, etc.
Inputting all TAPPS data and scores. You are either full on SID or provide info to person who has relationships with papers; or you manage and supervise coaches in their effort to provide info to papers in the manner and timeliness that you require.
Meet with coaches prior to or attend all district meetings that set schedules and rules, or which select all-district, etc. I was sole representative at one end-of-season district meeting which my coach could not attend and I had not seen one of their district games because I was coaching a different sport during the same season. Turned out OK, but not ideal.
Overseer of stats and manner in which stats are taken to assure quality.
Marketing of program. Includes outside and within school, two entirely different audiences and means of promotion.
Deal with Booster Club (if you have a good one) on concessions and gate, or otherwise responsible for all concessions and gate set up, personnel, stocking, collection, clean up, etc.
Deal with Booster Club (good or bad) on all money raising activities.
Go to most every game, game administrator, face of program. Talk to parents, listen to crowd, get the feel of your athletic program.
Oversee all attempts to make games entertaining, especially music issues. Huge at Christian schools. Try to infuse some fun into games.
Coach sport you are best at if you have any time whatsoever. Maybe go to middle school. May be more important than high school in getting kids and parents in line with school philosophy. I figure if I have the middle school in line, I have those parents and kids for 6 years. If I neglect the oversight at middle school, I may be trying to fix my mistake their whole high school careers.
Parent meetings. Just know they will be coming and the meetings will last almost an hour right in the middle of your day.
Student issues - -grades/behavior/home/boy-girl/anything really. I honestly believe my coaches and I have first responsibility for loving on or discipline of the athletes, regardless of where they are messing up. I tell the teachers to come to me whenever they have one of my kids who they feel are not giving effort in class, or are being disrespectful. I figure we can have a quick shot at correcting the problem before it goes to VP. Deal with admin and teachers on any issues we cannot fix at practice.
Spirit wear. I like to have final say on this. If you dont, some real junk can get out there with your name on it.
Scheduling, all aspects, quality of opponent, officials, home/road split, expense, conflict with school events avoidance, etc.
Scheduling, confirming, payment to refs. Deal with any bad ref issues with organization.
All dealings with governing body, including compliance with their rules, appeals that need to be written, enforcement of penalties imposed or corrective action that must be taken, etc.
Draw up all official protocol relating to key issues, concussions, emergency plan, travel/weather issues, etc.
Periodic report to Board or Directors.
Person in charge of putting out all fires, i.e, electricity goes out during game, AC stops working during game, one light standard goes out during game, big injury at game, officials do not show for game, one official late for game, anything out of the ordinary is on you to fix.
Coordination of/conflict mediator for all involved at game, including band, other teams band, cheerleaders, etc. Way bigger deal than you would think.
Supervisor of PE department for elementary and middle school. This could be another whole list 1/3 of what has already gone up. Much more detailed on the curriculum side (knowing specific daily lesson plans for each week of teaching), ordering/maintaining equipment, hiring/supervising/firing employees.
And last but not least, cheerleading. No, it is not a fine art; it is yours. And you better hope you have a good coach.
This is just off the top of my head after one year of AD work. This does not really even take into account just deep thought about the direction of or improvements to your program. Honestly, that is the one thing I am going to work hardest for this next year, finding time to just be able to think.