Asthma: This is the most likely diagnosis given the patient's history of wheezing episodes triggered by allergens (mowing grass, inhaling cigarette smoke) and her allergies to grass, tree, and ragweed pollen. Asthma is a chronic inflammatory disease of the airways that is characterized by recurring episodes of wheezing, coughing, chest tightness, and shortness of breath.

Allergic Rhinitis: While not directly causing cough, allergic rhinitis can lead to postnasal drip, which may irritate the throat and cause a chronic cough. The patient's allergies to pollen make this a plausible diagnosis.
Chronic Obstructive Pulmonary Disease (COPD): Although less likely in a non-smoker, COPD can cause chronic cough and wheezing. However, the patient's history of wheezing episodes in response to specific allergens leans more towards asthma.
Environmental Exacerbations of Asthma: Given her known allergies and asthma-like symptoms, exposures to allergens or irritants could exacerbate her condition, leading to chronic cough.

Lung Cancer: Although unlikely in a 44-year-old without a significant smoking history, lung cancer can present with chronic cough. It's crucial to consider this diagnosis to ensure timely intervention if present.
Tuberculosis (TB): Chronic cough is a hallmark of TB. Although less common in immunocompetent individuals without known exposure, it's a diagnosis that cannot be missed due to its public health implications and the need for specific treatment.
Sarcoidosis: This condition can cause chronic cough among other symptoms. It's an important diagnosis not to miss due to its potential for significant morbidity if left untreated.

Cystic Fibrosis: Typically diagnosed in childhood, cystic fibrosis can present in adults with chronic cough, among other symptoms. It's rare but should be considered in the differential diagnosis of chronic respiratory symptoms.
Eosinophilic Granulomatosis with Polyangiitis (Churg-Strauss Syndrome): A rare autoimmune condition that can cause asthma, chronic cough, and other systemic symptoms. It's an important consideration in patients with asthma and other systemic findings.
Bronchiectasis: A condition characterized by damaged, widened airways, which can lead to chronic cough. It might be considered if other diagnoses are ruled out and if there are findings suggestive of bronchiectasis on imaging.