For a 3-year-old who is coughing a lot but not running a fever, there are limited options for treatment. It is recommended to use a humidifier in the child's room at night to help relieve the cough. You can also try elevating the head of the bed with extra pillows to reduce nasal secretions that can cause coughing. There are over-the-counter products available, but their effectiveness is not scientifically proven. If the cough persists for more than 7 to 10 days without improvement, or if a fever lasts for 2 to 3 days, it is advisable to consult a doctor.