A FAN discount is a Foundation Account Number, typically obtained through an employer, educational institution, or other organization that AT&T has a relationship with. Once you apply it to your account, it will stay on your account and on any of the Mobile Share plans (original Mobile Share, Mobile Share Value, Mobile Share Advantage), it would apply to the data portion of your bill. On the now-gone DirecTV Unlimited Plan, it would apply to the $60 "data" portion. It does not apply to the new unlimited plans.
You can check with your work or school email address here or go to an AT&T store: https://www.wireless.att.com/business/enrollment/
I don't know the largest, but most people consider a 15-25% window very good. Most colleges and universities get 10-15% for students, faculty, and staff.
The only other discounts are the $25 toward TV service on the new Unlimited Choice plan (not sure how that actually shows up) and the occasional BOGO deals on phones (you're billed for the "free" phone every month for two years, and a bill credit for that amount is applied - this makes it free, but only if you stay through the two years).