  Usually when solving differential or difference equations via series
solutions one encounters divergent series in which the coefficients grow like a
factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the
size $q^{-\frac12 n(n-1)}$, in which $q\in(0,1)$ is fixed. Hence, the
divergence is much stronger, and one has to introduce alternative Borel and
Laplace transforms to make sense of these formal series. We will discuss
exponentially-improved asymptotics for the basic hypergeometric function
${}_2\phi_0$ and for solutions of the $q$-difference first Painlev\'e equation
$q{\rm P}_{\rm I}$. These are optimal truncated expansions, and re-expansions
in terms of new $q$-hyperterminant functions. The re-expansions do incorporate
the Stokes phenomena.
