Allergic Rhinitis: Given the patient's allergy to ragweed and the presence of a cough without fever or wheezing, allergic rhinitis is the most likely diagnosis. The cough is likely postnasal drip from the allergic reaction.

Postnasal Drip Syndrome: This condition often presents with a chronic cough due to the irritation of the upper airway by nasal secretions, which could be exacerbated by the patient's allergy to ragweed.
Asthma: Although the patient does not have wheezing, cough can be a sole symptom of asthma, especially in patients with allergies. The absence of wheezing does not rule out asthma entirely.
Chronic Sinusitis: This condition can cause a chronic cough due to postnasal drip and could be related to or exacerbated by the patient's allergies.

Lung Cancer: Although rare and less likely, a persistent cough for 5 weeks without an obvious cause like infection should prompt consideration of lung cancer, especially in smokers or those with a significant smoking history.
Tuberculosis (TB): A chronic cough without fever could be TB, especially if the patient has been exposed to someone with TB or has risk factors such as immunocompromised status.
Pertussis (Whooping Cough): This bacterial infection can cause a prolonged cough and is contagious. It's less likely but important to consider, especially if there's been exposure to someone with pertussis.

Eosinophilic Pneumonia: A rare condition characterized by an accumulation of eosinophils in the lungs, which could be related to allergies and presents with cough among other symptoms.
Hypersensitivity Pneumonitis: An allergic reaction to inhaled substances that can cause cough, among other respiratory symptoms. It's less common but could be considered in the context of allergies and exposure to specific antigens.
Cystic Fibrosis: Although typically diagnosed in childhood, some cases may not be diagnosed until adulthood. A persistent cough could be a symptom, but other signs like recurrent infections and malabsorption would also be present.