Acute Bronchitis: This is the most likely diagnosis given the presence of a cough with green sputum for 3 weeks without fever or shortness of breath (SOB). Acute bronchitis is a common condition characterized by inflammation of the bronchial tubes, often caused by a viral infection, but can also be due to bacterial infections. The green sputum indicates the presence of pus, which is a body response to infection.

Chronic Obstructive Pulmonary Disease (COPD) Exacerbation: If the patient has a history of COPD, an exacerbation could present with increased cough and sputum production, including green sputum, without necessarily having fever or SOB.
Asthma: Asthma can cause chronic cough, and if not well-controlled, could lead to the production of green sputum due to chronic inflammation and possible infection.
Pneumonia (Mild or Partially Treated): Although pneumonia typically presents with fever and possibly SOB, a mild case or one that has been partially treated could present with cough and green sputum without these other symptoms.

Lung Cancer: Although less likely, lung cancer can present with a chronic cough and could potentially produce green sputum if there is an associated infection. Given the potential lethality of missing this diagnosis, it's crucial to consider it, especially in smokers or those with a significant smoking history.
Tuberculosis (TB): TB can cause a chronic cough with sputum production, which may be green if there is a secondary bacterial infection. TB is a critical diagnosis not to miss due to its public health implications and the need for specific treatment.
Pulmonary Embolism: While typically presenting with acute onset of SOB, a pulmonary embolism could occasionally present atypically, especially if small, with cough and possibly sputum production without significant SOB.

Cystic Fibrosis: This genetic disorder leads to the production of thick, sticky mucus that can cause chronic cough and sputum production. It's rare in adults without a previous diagnosis but could be considered in those with a suggestive history.
Bronchiectasis: A condition characterized by damage to the airways, leading to chronic cough and sputum production. It's less common but should be considered in patients with a history suggestive of recurrent infections or specific risk factors.
Aspiration Pneumonia: If the patient has a history of dysphagia or other risk factors for aspiration, aspiration pneumonia could be a rare cause of chronic cough with green sputum, especially if the aspiration is chronic and low-grade.