If your goal is to read Demazure and Gabriel's book, then (as I explained in the comments) your question is based on false premises and the solution is to read the definitions carefully.
But let me address your question as written, since it will illuminate why Demazure and Gabriel use the definitions they do.
First, as you observe, the category of elements of an arbitrary functor $F : \textbf{CRing} \to \textbf{Set}$ is not always small.
Actually, it is almost never small, because $\textbf{CRing}$ itself is not small: as soon as $F (A)$ is non-empty for all rings $A$, then the category of elements of $F$ will be at least as big as $\textbf{CRing}$.
This is not actually fatal for the problem at hand (though it does introduce many complications).
It sometimes happen that the functor $F$ you are interested in is a colimit of a small diagram of representable functors, i.e. there is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F (B) \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, B)$ naturally in $B$.
In that case, you can compute the colimit $\left| F \right|$ you seek as $\varinjlim_\mathcal{I} \operatorname{Spec} A$.
The biggest complication is that $\left| F \right|$ is not well defined for arbitrary $F$, so you only get a partially defined functor $\left| - \right|$.
If you try to restrict to a full subcategory of functors $\textbf{CRing} \to \textbf{Set}$ on which $\left| - \right|$ is well defined everywhere, you then have the complication that the putative right adjoint may not have image contained in that subcategory.
I am not aware of any good way to resolve this dilemma; I think you have no choice but to settle for a partially defined adjoint.
Now for some good news: there is a clean necessary and sufficient condition for a functor $F : \textbf{CRing} \to \textbf{Set}$ to be a colimit of a small diagram of representable functors.
Definition.
Let $\kappa$ be an infinite regular cardinal.
A $\kappa$-accessible functor is a functor that preserves $\kappa$-filtered colimits.
Proposition.
Let $F : \textbf{CRing} \to \textbf{Set}$ be a functor.
The following are equivalent:
- $F$ is $\kappa$-accessible.
- $F$ is the left Kan extension of a functor $\textbf{CRing}_\kappa \to \textbf{Set}$ along the inclusion $\textbf{CRing}_\kappa \hookrightarrow \textbf{CRing}$, where $\textbf{CRing}_\kappa$ is the full subcategory of $\kappa$-presentable rings (i.e. rings presentable by $< \kappa$ generators and $< \kappa$ relations).
- There is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, -)$ and, for each $i$ in $\mathcal{I}$, $A (i)$ is a $\kappa$-presentable ring.
The functor $R \mapsto R^{\oplus \mathbb{N}}$ you mention is easily seen to preserve filtered colimits (i.e. be an $\aleph_0$-accessible functor).
It is just as easy to see that it is the colimit of a small (indeed, countable!) diagram of representable functors, namely,
$$\textbf{CRing} (\mathbb{Z}, -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1], -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1, x_2], -) \longrightarrow \cdots$$
where the maps are the ones induced by the homomorphisms $\mathbb{Z} [x_1, \ldots, x_n, x_{n+1}] \to \mathbb{Z} [x_1, \ldots, x_n]$ that send $x_i$ to $x_i$ for $1 \le i \le n$ and $x_{n+1}$ to $0$.
Thus, the geometric realisation of $R \mapsto R^{\oplus \mathbb{N}}$ is the colimit $\varinjlim_n \mathbb{A}^n$.
I suppose I owe you an example of a functor $\textbf{CRing} \to \textbf{Set}$ that is not accessible.
Choose an ordinal-indexed sequence of fields, $K_\alpha$, such that $K_\alpha$ is strictly smaller in cardinality than $K_\beta$ whenever $\alpha < \beta$.
Let $F (R) = \coprod_{\alpha} \textbf{CRing} (K_\alpha, R)$ for non-zero rings $R$ and let $F (\{ 0 \}) = 1$.
Since any ring homomorphism $K_\alpha \to R$ is injective when $R$ is non-zero, $\textbf{CRing} (K_\alpha, R)$ is empty for sufficiently large $\alpha$, so $F (R)$ is indeed a set.
On the other hand, it is clear that $F$ cannot be the left Kan extension of any functor $\textbf{CRing}_\kappa \to \textbf{Set}$: if it were, it would be impossible to distinguish between this $F$ and the one where we cut off the disjoint union at some ordinal $\beta$ such that $K_\beta$ is not $\kappa$-presentable.
Best Answer
The forgetful functor from locally ringed spaces to topological spaces is actually a left adjoint, not a right adjoint; in particular, it preserves colimits. You can easily see it is not a right adjoint because it does not preserve limits: the terminal locally ringed space is $\operatorname{Spec}(\mathbb{Z})$, but its underlying space is not the terminal topological space (a singleton).
The right adjoint to the forgetful functor is fairly complicated. Roughly speaking, it takes a topological space $X$, equips it with the constant sheaf $\mathbb{Z}$ (which gives the right adjoint if you were just taking ringed spaces, not locally ringed spaces), and then "expands out" every point of $X$ into new points for each prime ideal in the stalk in order to get a locally ringed space. For more discussion and references, see this nice answer.
For the specific example you are interested in, you can just easily directly construct the colimit. Namely, let $X$ be a singleton equipped with the sheaf of rings whose global sections are $k[[x]]$. This is a locally ringed space, and it has compatible maps from your diagram. To see that it is the colimit, let $Z$ be any locally ringed space with compatible maps from your diagram. All these maps must have image that is the same single point $z\in Z$, and we get a compatible system of maps $O_{Z,z}\to k[x]/x^n$ for each $n$. These induce a map $O_{Z,z}\to k[[x]]$, and this gives a morphism of locally ringed spaces $X\to Z$ that sends the point of $X$ to $z$ and sends a section of $O_Z$ in a neighborhood of $z$ to its image in $k[[x]]$. Moreover, it is easy to see this is the unique such morphism compatible with the maps from the diagram.
(More generally, you can explicitly construct colimits of locally ringed spaces by taking the colimit of the underlying topological space and then equipping it with essentially the only obvious sheaf of rings you can define. Of course, the details are much more complicated to check in general than in your example where everything is just a singleton.)