  In this paper, we examine the boundary behaviour of the generic power series
$f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense of
Baire category. Notably, we prove that for any open subset $U$ of the unit disk
$D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of
$\CC$. As it is demonstrated, this conclusion does not necessarily hold for
arbitrary open sets accumulating to the unit circle. To complement these
results, a characterization of coefficient sets having this property is given.
