  Domains that are increasing union of balls (up to biholomorphism) and on
which the Kobayashi metric vanishes identically arise inexorably in complex
analysis. In this article we show that in higher dimensions these domains have
infinite volume and the Bergman spaces of these domains are trivial. As a
consequence they fail to be strictly pseudo-convex at each of their boundary
points although these domains are pseudo-convex by definition. These domains
can be of different types and one of them is Short $\mathbb{C}^k$'s. In pursuit
of identifying the Runge Short $\mathbb{C}^k$'s (up to biholomorphism), we
introduce a special class of Short $\mathbb{C}^k$'s, called Loewner Short
$\mathbb{C}^k$'s. These are those Short $\mathbb{C}^k$'s which can be exhausted
in a continuous manner by a strictly increasing parametrized family of open
sets, each of which is biholomrphically equivalent to the unit ball and
therefore, they are Runge up to biholomorphism. Although, the question of
whether all Short $\mathbb{C}^k$'s are Runge (up to biholomorphism), or whether
all Short $\mathbb{C}^k$'s are Loewner remains unsettled, we show that the
typical Short $\mathbb{C}^k$'s are Loewner. In the final section, we construct
a bunch of non-autonomous basins of attraction, which serve as interesting
examples of Short $\mathbb{C}^2$'s.
