Is there a notion of "schemeification" analogous to that of sheafification of a presheaf? So this may seem like an odd question, but hear me out. In the Stacks Project, tag 01I4, we find that not only does the category of affine schemes live inside the category of locally ringed spaces, but that limits of affine schemes can be computed as limits in the ambient category of locally ringed spaces. In other words, the inclusion functor commutes with these limits. 
I am also aware that the reason we have sheafification of a presheaf is similar. The inclusion of the category of sheaves on a space is a full subcategory of the category of presheaves on that space. Moreover, it commutes with limits and a certain smallness condition is satisfied that allows us to deduce that the adjoint functor theory is satisfied and hence the inclusion functor has a left adjoint, which we call the sheafification functor.
Is it true that the inclusion of affine schemes into the category of locally ringed spaces also admits a left adjoint which we might call schemeification? If so, what does it look like? How does it turn a locally ringed space into an (affine?) scheme? If not, what fails that we can't apply the adjoint functor theorem? 
 A: There is an "affine schemification", a left adjoint to the inclusion of affine schemes into locally ringed spaces.  Indeed, given a locally ringed space $X$, the left adjoint just sends $X$ to $\operatorname{Spec} \mathcal{O}_X(X)$.  To sketch the proof, there is a canonical map $X\to\operatorname{Spec}\mathcal{O}_X(X)$, since each point $p\in X$ can be sent to the prime ideal of global functions which are in the maximal ideal of $\mathcal{O}_{X,p}$.  You can then verify that this actually naturally can be made into a morphism of locally ringed spaces and is initial among all morphisms from $X$ to affine schemes.  See Theorem V.3.5 of Peter Johnstone's Stone spaces for more details.
However, there is not a "schemeification" if you allow all schemes and not just affine schemes.  The problem is that schemes do not have arbitrary limits.  For instance, an infinite product of non-affine schemes typically does not exist; see this MO question.  On the other hand, the category of locally ringed spaces has all limits (this is not obvious; see Corollary 5 of this paper).  It follows that there can be no schemeification functor, since a reflective subcategory must be closed under limits.  Explicitly, if you have a diagram of schemes that has no limit in schemes, then its limit in locally ringed spaces has no schemeification, since its schemeification would be a limit in schemes.
For some related discussion, see this very nice answer by Martin Brandenburg.
