A quick answer for now, which I might add more to later. Recall that for every adjunction $F \dashv G: C \to D$ there is the notion of "fixed point" of the adjunction which has two faces: either it is an object $c$ of $C$ for which the counit $\epsilon_c: FGc \to c$ is an isomorphism, or it is an object $d$ of $D$ for which the unit $\eta_d: d \to GFd$. The adjunction $F \dashv G$ then induces an adjoint equivalence between the full subcategories $\text{Fix}_{FG}(C)$ and $\text{Fix}_{GF}(D)$, and thus we identify these categories. In the case of the Isbell conjugation adjunction for a small category $X$, the category of fixed points is often called the *Isbell envelope* or *Isbell completion*; let's denote it $I(C)$. 

In the special case where $C$ is a preorder, i.e., a $\mathbf{2}$-enriched category, $I(C)$ is a preorder which goes by a more famous name: the Dedekind-MacNeille completion. You can find some discussion of this in the MO-answer <a href="http://mathoverflow.net/a/59295/2926">here</a>. In this case $I(C)$ *is* both complete and cocomplete (i.e. admits small limits and colimits). 

In other cases, the Isbell completion need not be $\mathcal{V}$-complete/cocomplete, as you correctly surmise (see comments below the MO-answer I just mentioned), but it can be quite a bit larger (and certainly larger than the Cauchy completion). It's maybe best to point to some examples. For the case where $\mathcal{V}$ is the monoidal closed category $([0, \infty), \geq, +)$, i.e., where $\mathcal{V}$-categories are metric spaces in the sense of Lawvere, Simon Willerton has an interesting article which connects the Isbell completion to the tight span of metric spaces; you can find a Caf&eacute; discussion <a href="https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html">here</a> where some examples are computed. 

Isbell completions also figure in "<a href="https://arxiv.org/pdf/0802.2225v2.pdf">comparative smootheology</a>", but I'd need to study and think more before commenting on that. You can also find more references to the Isbell completion in the nLab; as I said, I may come back and add more.