The story explaining why an immortal should look like a crippled beggar is the best known of the group and is told with the timing of a fable.
Li was an adept whose spirit could leave his body and travel, and on one occasion he was summoned, in some versions by Laozi himself, to a meeting on a distant mountain. He told his disciple to keep watch over the body and to burn it if he had not returned within seven days, since after that it would be past use. On the sixth day word came that the disciple's mother was dying. The young man waited as long as he dared, then burned the body and went.
The spirit returned on the seventh day to nothing, and had to take the only vacant body within reach: that of a beggar who had just died of starvation at the roadside, lame, one-eyed and filthy. The accounts have him about to abandon it in disgust and being told to keep it, and in several versions Laozi gives him a gold band to bind the wild hair and an iron crutch to walk with.
The tradition draws the moral without much subtlety and it is a good one anyway. The people in the stories who take him for a beggar and treat him accordingly are the ones shown to be mistaken, and the immortal in the ruined body is a standing test of whether anybody is paying attention.