Building state-of-the-art (SOTA) predictive models for drug discovery requires expensive search over tools, architectures, and training strategies. Current LLM-based agents can find SOTA solutions through extensive trial and error, but they do not retain the experience accumulated along the way and therefore pay the full search cost on every new task. We propose \method (Self-evolving Agent Experience), a framework that accumulates and reuses experience across tasks to build SOTA drug discovery models efficiently. \method maintains a cross-task memory of verified skills, statistical evidence about effective strategies, and a record of recurring errors and their fixes. In some cases, \method transfers a working solution directly without test-time search. In 33 molecular property prediction tasks, \method ranks first among nine SOTA agents in a single-task setting. With memory accumulated from 16 smaller tasks, \method achieves an averaged normalized score of 0.935 on 17 held-out tasks in a cross-task evaluation setting and outperforms all baseline agents by 10-30\% in a zero-test-time search regime. In summary, our work shows the advantage of cross-task memory for efficient SOTA model development in drug discovery.