How many edges can a unipathic graph have?
A unipathic graph can have $\Theta(n^2)$ edges. There's a well-known kind of graph that's unipathic and has $n^2/4$ edges.


   Consider a complete bipartite graph, with oriented edges $\forall (i,j) \in [1,m]^2, a_i \rightarrow b_j$. This graph is unipathic and has no cycle: all its paths have length $1$. It has $2m$ vertices and $m^2$ edges.


 (Follow-up question: is this ratio maximal? Probably not, but I don't have another example. This example is maximal in the sense that any one edge that you add between existing nodes will break the unipathic property.)