Which process is most responsible for the increase in early Earth's atmospheric oxygen levels?

The answer is photosynthesis.

Photosynthesis occurs in the leaves of green plants. During photosynthesis, carbon dioxide and water are converted into using the energy of sunlight to glucose and oxygen. After great evolution of plants, the photosynthetic rate was increased. More plants mean more oxygen produced in the process of photosynthesis. Thus, photosynthesis is most responsible for the increase in early Earth’s atmospheric oxygen levels.