I've been told that Sai Baba encouraged his followers to be independent. Is there a story in the Satcharitra that illustrates this teaching?

Yes, Sai Baba often guided his devotees towards self-reliance. A clear example of this is found in the story of Narayan Motiram Jaani, detailed in Chapter 33. Narayanrao was employed as a servant for another devotee. When he came for Baba's darshan, Baba specifically instructed him that his time of servitude was over and that he should start his own independent business. Trusting Baba's words, Narayanrao established an enterprise named "Anandashram." This venture flourished, bringing him peace and prosperity, and his faith in Baba's foresight and guidance became firm and unshakeable. This leela beautifully demonstrates Baba's desire to see His devotees stand on their own feet.