One of the new examples for the third edition of Bayesian Data Analysis is a spell-checking story. Here it is (just start at 2/3 down on the first page, with “Spelling correction”).
I like this example—it demonstrates the Bayesian algebra, also gives a sense of the way that probability models (both “likelihood” and “prior”) are constructed from existing assumptions and data. The models aren’t just specified as a mathematical exercise, they represent some statement about reality. And the problem is close enough to our experience that we can consider ways in which the model can be criticized and improved, all in a simple example that has only three possibilities.