What is the infinite dual space of a vector space?

Question

Given an infinite dimensional vector space, $V$, its dual space, $V^*$, is larger than $V$ and its double dual even bigger (usually). If you keep on going with the triple and quadruple dual, what happens? Taking the limit of the system where $V_0=V$ and $V_i=V_{i-1}^{**}$ with the morphisms being the canonical injections from a space into its double dual, what would the direct limit be? And also correct me if this isn’t the proper way to formulate this, category theory isn’t my strong suit.