Is it possible for a child-site of a multisite to be a multisite itself?

No, wordpress finds out the proper place in the DB to look for a sites data by looking at the first directory name and there is no way to have a “tree” style lookup.
You might be able to hack around it, and implement a tree style lookup by having a more complex mapping function of your own but you will be probably in uncharted territory.

side note: the multi network feature of wordpress is part of it for more then 5 years. If after all this time no one bothered to add a UI for it in the core then it is a good indication that most developers don’t think it is a better solution then installing a new wordpress.