In the state of Alaska, if you are a salaried employee, is there a required amount of days off per week, per month, per quarter, etc.? Or does your employer even have to give you any days off?

I could not find any reference to a "one day's rest in seven" provision in Alaska law. Accordingly, whether you are "salaried" or hourly (which are merely pay methods) or exempt or nonexempt (from the minimum wage and overtime provisions of the Fair Labor Standards Act), the employer is not required to provide any specific days off or days off after working XX many days in a row.

This is not unusual. Most states do not have such regulations.