Every cat on the street gets greeted, whiskers get drawn on everything, and there has been a campaign for a kitten since your child could talk. Pick the one that suits your child's age and the mood they're in, and that's the whole trail done. There's a toddler's bowl-of-milk book here and a cat martial-arts novel and a warrior clan saga, and any single one of them feeds the obsession on its own.
Any one of them satisfies a cat-mad child completely. What varies is age and flavor. For the very small there's a kitten chasing the moon, a walk through a dozen different animals' eyes, and a blue cat singing about his ruined shoes. Around five or six there's a 1928 folk tale with a chant they'll never stop repeating, and a cat who goes bad in alphabetical order. For an older child who wants a real story with a cat at the center, there's a pedigree house cat learning a secret martial art, and a clan saga big enough to eat a year of bedtimes. The gap between the youngest and oldest books here is enormous, which is the point. Pick by the child in front of you.