Osteochondroma: This is the most likely diagnosis given the description of the bony growths arising from the posterior aspect of the distal femoral shaft. Osteochondromas are the most common type of benign bone tumor and typically present as a bony growth on the surface of the bone, often near the growth plate.

Enchondroma: Although less common in this location, enchondromas are benign cartilaginous tumors that can present as bony growths. They are more commonly found in the short tubular bones of the hands and feet but can occur in long bones.
Bone island (Enostosis): A bone island is a benign growth of compact bone within the cancellous bone. While it can appear as a bony growth, it is typically denser than the surrounding bone and might not exactly match the description given.
Ossifying hematoma or myositis ossificans: These conditions involve the formation of bone within soft tissues (muscle) following trauma. They could potentially present as bony growths near the femur but would typically have a history of trauma.

Chondrosarcoma: Although less likely, chondrosarcoma is a malignant tumor that can present similarly to an osteochondroma. It is crucial to distinguish between benign and malignant processes, as the treatment and prognosis are vastly different.
Osteosarcoma: This is the most common primary malignant bone tumor, which can present as a bony growth. Early diagnosis is critical for treatment and survival.
Metastatic bone disease: In adults, metastatic lesions to the bone are more common than primary bone tumors. These can present as bony growths and would require immediate attention due to their implications for the underlying primary malignancy.

Turret exostosis: A rare condition characterized by a bony growth that can occur after a traumatic event, leading to the formation of an osteochondroma-like lesion.
Subungual exostosis (if the growth were to be located near the knee joint and involved the patella or tibial tubercle): Although typically described in the context of the toes, a similar process could theoretically occur near other joints, presenting as a bony growth.
Melorheostosis: A rare bone disorder characterized by an excessive cortical bone formation, which can present as a bony growth or sclerosis along the surface of the bone.