Should I see a doctor or leave it?

A persistent cough does need to be treated by a doctor. A cough is the body's reaction to irritants in the lungs and airway. It is your body's way of getting the "gunk" out. Gray or brown phlegm can be a sign of infection, and could be bacterial, which would merit an antibiotic. Your doctor can draw blood to check for a viral vs. bacterial infection, and may order an xray to be sure that you do not have pneumonia.