What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be interested in asymptotics or lower bounds. For more about median graphs see the survey by Klavžar and Mulder:

- Sandi Klavžar, Henry Martyn Mulder,
*Median graphs: characterizations, location theory and related structures*, J. Combin. Math. Combin.**30**(1999) 103–127. (preprint)

Imrich et al. claimed to have shown (at least "intuitively") that there are as many median graphs as there are triangle-free graphs. Their argument involves composing a sequence of injections. Since the sets involved are infinite, this is unfortunately not rigorous enough to conclude anything about the actual numbers.

- Wilfried Imrich, Sandi Klavžar, Henry Martyn Mulder,
*Median graphs and triangle-free graphs*, SIAM J. Discrete Math.**12**(1999) 111–118. doi:10.1137/S0895480197323494 (preprint)

Given the apparent connection with triangle-free graphs, I would also be interested in asymptotics or bounds for the number of $n$-vertex triangle-free graphs, for which OEIS has the following relevant sequences: A006785, A024607.

[In an effort to make the question more self-contained, I append the definition of median graph from the Wikipedia link: In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices $a$, $b$, and $c$ have a unique median: a vertex $m(a,b,c)$ that belongs to shortest paths between each pair of $a$, $b$, and $c$.]