I am looking for both upper bounds and lower bounds on the number of maximal intersecting families (intersecting families of size $2^{n-1}$)
A trivial lower bound is $n+1$ (by considering $\mathcal{A}_i = \{A \in \mathcal{P}(n): i \in A\}$ and $\mathcal{A} = \{A \in \mathcal{P}(n): |A| \geq \lceil n/2 \rceil\}$.
Noting that a maxinmal intersecting family must be an upset yields an asymptotic upper bound of $n2^{n^2}$. (some level set provides a basis for your intersecting family)
My question is; can these bounds be improved?