Final answer:
The President of the United States is the authority that can grant HYT for an E₁-E₂ services member to re-enlist or extend their service.
Step-by-step explanation:
The authority that can grant HYT (High Year Tenure) for an E₁-E₂ services member to re-enlist or extend their service is the President of the United States. The President of the United States is the authority that can grant HYT for an E₁-E₂ services member to re-enlist or extend their service.
According to Article II, Section 2 of the United States Constitution, the President is designated as the Commander in Chief of the Army and Navy of the United States. This gives the President the power to make decisions regarding the armed forces, including granting HYT for service members.