What is the maximum number of shortest paths between any pair of vertices in a chordal graph?
It can be exponential in $n/2$.  The following graph has $2^{(n/2) - 1}$ shortest paths between the endpoints:

This graph has $2^{(14/2)-1} = 2^6 = 64$ different shortest path, each corresponding to a selection of "ups" and "downs".