[Distilled from this question, with less informality.]
What are some things a new Cocoa programmer should write as sample projects, outside of the hand-holding of a tutorial?
Let's have one project idea per answer. If you have multiple ideas to suggest, post multiple answers.
It'll help for you to also list what Cocoa concepts your project idea will teach/reinforce.
Also, if your answer addresses Cocoa Touch- or Cocoa (Mac)-specific concepts, please indicate that. This question is for both Cocoa and Cocoa Touch.