Hospital/clinic practice: People who don't read their health insurance policies and can't believe it when you can't get them set up for X, Y, and Z when their policies don't cover it. After that, it's the health insurance carriers who deny things that are in the policy and waste your time with prolonged appeals. PIA.
If you're in the work comp arena, it's a toss-up between employers who won't provide duty for transition to work after injury and the adjusters who think they know all about medical care and won't approve something that is absolutely essential for an injured worker to recover.
If you're in private practice, it's getting your invoices paid. Although you can tell them how much time you'll need and there are good published resources on estimating time required for case management, they are always astounded by the bill for the first month when you have done all the leg work and set up and contacts.