LLCs must have at least one member with ownership interest, otherwise it (or all its assets) would be abandoned property by state law
That is, whether there's anything that stops a sequence of share sales ending up in a situation where a set of companies between them own all the shares of companies in the set.
(If this ever happened, presumably the last human to sell a share to one of the companies was making a mistake, but humans make mistakes all the time.)
I mean, normally if you wanted to stop this sort of thing you'd do something like introduce a hierarchy of company types and say type N companies may only own shares in companies of type N+1 or higher. But I've never heard of such a rule.