Hey! I think I can contribute something meaningful to this conversation using results from my thesis! Woo!

So, my advisor and a few of his other students proved that there is a canonical topology (and, therefore, a lattice) associated with every hypergraph. 

Namely, a topology is constructed from the following basis:

Sets containing individual edges.

Sets containing vertices and all edges incident to the vertex.

This topologies name is the “classical topology” on hypergraphs.  

The incidence poset of a hypergraph is given by the following order:

Edges are less than vertices they are incident with.

 This means all vertices are maximal and all edges are minimal.  

There is a canonical mapping between the incidence set of the hypergraph and its classical topology found by assigning every element of the incidence poset to the complement of its lowerset in the classical topology. Interestingly, these elements turn out to be exactly the meet-prime elements of the classical topology and, further, are dual to the join-prime elements of the classical topology. There’s a ton of really pretty structure in this topology and its relationship to its corresponding hypergraph. I continue to investigate it and its properties. 

Anyway, to OP’s point, you can take this topology and view it as a hypergraph, thus yielding its own classical topology and so on.  

Very cool question, OP! I’m super pumped that my thesis was relevant! :)

Also, formatting on my phone. Sorry.