Which part of the brain is responsible for controlling the breathing process?

The medulla oblongata is the correct choice because it plays a crucial role in the autonomic control of vital functions, including respiration. It houses the respiratory control centers that regulate the rhythm, depth, and rate of breathing. When the body requires more oxygen, such as during exercise, the medulla oblongata responds by adjusting the breathing rate accordingly.

In contrast, the hippocampus is primarily involved in memory formation and spatial navigation, the cerebellum is essential for coordination and balance, and the thalamus acts as a relay station for sensory information. None of these regions are directly responsible for controlling the respiratory process in the way the medulla oblongata is. Thus, the key role of the medulla oblongata in respiratory regulation establishes it as the correct answer.