Here is the thing - small business owners are always struggling to get things done and reduce costs. That provides you a great opportunity! If you are already working there, you probably see little ways that things could be better - a process could be more efficient, there is maybe some waste occuring with product somewhere, or there are tasks that need to be done and you would like to do them.
The trick is to bring up these items without making them seem like crititcisms because small business owners often see their business as their baby! I would recommend writing down some of the opportunities you see for improvement and also, for each item, write down a solution that would save the business owner some money or time, but not cost them more money.
Then, find a time when the business owner is having a GREAT day. I mean, when they are in a fabulous mood. Tell them how much you love working there and that you would really like to see yourself growing more there, taking on more responsibility, etc. Ask them if you can set up a time to talk to them about your future and how you can add more value to the business. Be confident and look them in the eye when you are talking to them, so that they can see you sincerity. Be prepared for them to say "Right now sounds good" or "Great, let's put something on the calendar". Either way, you will already be prepared with your notes and ideas.
If the business owner is resistent to you growing more in your role, then maybe this isn't the right place for you right now. Look at other companies in your area that are similar and see if they are hiring. Good luck to you!