Recursive relationship database
As well as management hierarchy, large businesses sometimes have divisional structures (country, area, region, city, branch) of fixed or variable depth. Geographic classification of places more generally could be a useful example too. Any taxonomy (books, living creatures, ...) or most other classification systems, depending on any additional constraints you have been given.

Or family trees. If you ignore complications where parentage is not known or legally changes, both the maternal and paternal lines are potentially infinite tree structures. If you consider them both together rather than as two separate structures connecting the same set of nodes, they form a directed graph which might also be within your spec.

If graphs generally are allowed then that opens up the field even more: for instance places connected by roads, more general relationships between people, or links between articles on something like Wikipedia or the web more generally. Heading back to the more specific (and back to a tree/forest structure) the posts on forums like Reddit of HackerNews.

Recursive structures can appear everywhere that you model things connected or related to each other.

If you are looking for existing data sets to analyse rather than things to model, then there are a great many out there to be found. /r/datasets might be a good place to start, for instance this post links to a dump of HN data.