I'd think of it as the set of sequences of elements of $V$, i.e.
$$ V^\mathbb{N} = \prod_{n=1}^\infty V.$$
But I don't think there is a really nice description. In the case where $V$ is finite-dimensional, it'd just be $K^\mathbb{N}$ in a clever (and non-canonically isomorphic) disguise, but if $V$ were to be infinite-dimensional, $V^\mathbb{N}$ could be rather ugly. E.g. $\ell_1^\mathbb{N}$: sequences of absolutely summable sequences. Then a sequence of sequences would converge to a sequence iff it converged coordinatewise. Complicated and very confusing.
Following Munkres' "Topology", I prefer to use $V^\infty$ for set of sequences in $V$ for which only finitely many elements are nonzero, i.e.
$$V^\infty = \bigoplus_{n=1}^\infty V = \left\lbrace (v_n) \in V^\mathbb{N} \;{\large\mid}\; \text{$v_n = 0$ from some index onwards}\right\rbrace. $$