Beta-Thalassemia Major: This condition is characterized by severe reduction in the production of the beta-globin chains of hemoglobin, leading to elevated levels of hemoglobin A2 and F (fetal hemoglobin) as the body attempts to compensate for the lack of normal hemoglobin. The age of presentation and the specific elevation of these hemoglobin fractions make this a highly likely diagnosis.

Beta-Thalassemia Minor: While typically less severe than beta-thalassemia major, some forms of beta-thalassemia minor can present with elevated hemoglobin A2 and sometimes elevated fetal hemoglobin, especially in early infancy.
HbS/Beta-Thalassemia: A compound heterozygous state for sickle cell disease and beta-thalassemia, which can also lead to elevated levels of hemoglobin A2 and F due to the beta-thalassemia component.

Alpha-Thalassemia Major (Hb Bart's Hydrops Fetalis): Although typically diagnosed prenatally or at birth due to severe anemia and fetal hydrops, it's crucial not to miss this diagnosis. However, the presentation with elevated Hb A2 and F is less typical for alpha-thalassemia major.
Other Hemoglobinopathies: Certain other hemoglobinopathies, like HbE/beta-thalassemia, can present with similar laboratory findings and should be considered to avoid missing a significant diagnosis.

Delta-Beta-Thalassemia: A rare condition affecting the production of both delta and beta chains, leading to elevated fetal hemoglobin and sometimes elevated Hb A2.
Hereditary Persistence of Fetal Hemoglobin (HPFH): A benign condition where fetal hemoglobin production continues into adulthood, which could be considered in the differential but is less likely given the clinical context of a 7-month-old with significant elevations.
Other Rare Hemoglobinopathies: There are numerous rare hemoglobinopathies that could potentially present with elevated Hb A2 and F, emphasizing the importance of comprehensive diagnostic testing.