Final answer:
A 15-year-old male with darkening skin of the axillae, a high BMI, and family history is most likely experiencing Acanthosis nigricans, which is often linked with insulin resistance.
Step-by-step explanation:
The most likely diagnosis for a 15-year-old white male who is presenting with complaints of darkening skin of both axillae, with a familial history of a similar condition, and a BMI of greater than 98, is Acanthosis nigricans. Acanthosis nigricans is a skin condition characterized by dark, velvety discoloration in body folds and creases and is commonly associated with obesity. This condition often signals insulin resistance, which is a risk factor for developing type 2 diabetes. Other conditions such as vitiligo, tinea versicolor, and melasma have distinguishing features that do not match this presentation. Vitiligo is characterized by lighter colored patches of skin due to depigmentation, while tinea versicolor would present with lighter or darker patches that may also be scaly, and melasma commonly involves dark patches typically on the face and is more common in women.