My boss wants me to list a few career Aspirations. I am currently an admin assistant. What should I say? My company is a insurance company.

Perhaps you might aspire to become an administrator and have your own assistant someday. Maybe you'd like to get your license to sell insurance. Maybe you would someday like to have your own insurance agency. You may be more interested in general office management or something in the field of technology. He's trying to find out what direction you'd like to go in. If you want to be an administrative assistant for the rest of your life, that's okay too, but you'll get to a point where you'll stop learning and growing and lose your potential to make more money. It's a tough question, but one we all have to face; "what do I want to be when I grow up?".

What should my career aspirations be for a Supervisor job

Do you want to be admin assistant for your whole career, or not. If not what do you want to do in the future. Do you want to be an HR, do you want to sell insurance, or do you want to be accountant. Nobody can answer this question except for you because nobody knows what you want.