Exploration is crucial for enabling legged robots to learn agile locomotion behaviors that can overcome diverse obstacles. However, such exploration is inherently challenging, and we often rely on extensive reward engineering, expert demonstrations, or curriculum learning - all of which limit generalizability. In this work, we propose Skill Discovery as Exploration (SDAX), a novel learning framework that significantly reduces human engineering effort. SDAX leverages unsupervised skill discovery to autonomously acquire a diverse repertoire of skills for overcoming obstacles. To dynamically regulate the level of exploration during training, SDAX employs a bi-level optimization process that autonomously adjusts the degree of exploration. We demonstrate that SDAX enables quadrupedal robots to acquire highly agile behaviors including crawling, climbing, leaping, and executing complex maneuvers such as jumping off vertical walls. Finally, we deploy the learned policy on real hardware, validating its successful transfer to the real world.