Number of simple paths between two vertices in a DAG
Every simple path is uniquely determined by the subset of vertices that it passes through: if you topologically order the DAG (arbitrarily) then a path through any subset of vertices must go through those vertices in the same order given by the topological order. So (since $s$ and $t$ must always be included) there are at most $2^{n-2}$ paths.

This bound is tight, for a complete DAG (one in which every pair of vertices is connected by an edge, oriented consistently with a single topological order).