A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. Can you give an example of complex $2$-hyperbolic manifold?
A non-trivial example is $CP^3$ minus 5 hyperplanes in general position. According to a theorem of Borel every holomorphic image of $C$ in this manifold is contained in a plane. (And there are finitely many of these planes). So every image of $C^2$ is also contained in a plane. And it is easy to see that the images can "fill" several planes. So $k=2$ according to your definition.