Suppose I have a holomorphic function $f$ in a domain $\Omega$ with natural boundary $\partial \Omega.$ Let $p \in \partial \Omega.$ Is it true that there is some analogue of Picard's little theorem - that is, by choosing an appropriate sequence $x_1, \dotsc, x_n, \dotsc \in \Omega$ converging to $p$ we can have the limit of $f(x_i)$ be (almost) any complex number?

**EDIT** As divined by Noam Elkies, this was inspired by the recent discussion of $\sum_{i=0}^n z^{i^2},$ and the fact that its zeros seem to cluster near its natural boundary. Indeed, consider the function (a slight variant of that occurring in both answers)
$$
\sum_{i=0}^n \frac{z^{2^i-1}}{(i+1)^2}.
$$

Here is the graph of its zeros($n=10$):

This would tend to imply that you don't have to work very hard to tend to zero when approaching the boundary, and I assume that zero is not a particularly exceptional value. Indeed, if you plot the $1$s of the function (preimages of the value $1$), you get an identical plot:

Which would tend to indicate that my conjecture has at least a grain of truth in it.