Let $X\subset \mathbb{C}^{n}$ be a domain. You can assume that it is nice (e.g. bounded convex balanced ). Let $\{x_n\}$ be a sequence of points that does not have a limit point in $X$.

Let $D$ be the (unit) disc on the plane.

Is there a holomorphic $\varphi:D\to X$ such that $\varphi(D)$ contains an infinite number of $x_n$?

Of course there are further questions: can we catch all the points? Or maybe all but finite number?