The point of self-improvement is not to reach some ideal self, but not to stay in the same place or get worse. So suppose I hadn't started to write music, hadn't taken piano lessons or sung in the choir. I would be the same person, but without whatever growth I achieved from going into music more seriously. I didn't need to learn to read Italian: I would have been fine without doing so. I could give up crossword puzzles and still have a satisfying life, without trying to do them faster and faster every day.
The idea that I need to find new research projects. I could easily just coast the rest of my career, and teach things I have already learned rather than come into the classroom with things I have learned in the past few years, as I like to do.
Without self-improvement, though, the world narrows rather than expanding. I would find it difficult to imagine being in a teaching situation in which I couldn't be a learner myself. It would go stale pretty quickly, and I think the students would notice too.