Background
Let $\mathcal{F} : \textbf{CRing} \to \textbf{Set}$ be a functor and denote by $\textbf{P}_\mathcal{F}$ the category of points of $\mathcal{F}$ whose objects are pairs $(R , \rho)$ where $R$ is a ring and $\rho : h^R \to \mathcal{F}$ is a morphism (here $h^R := \operatorname{Hom}_{\textbf{CRing}}(R , -) \in [\textbf{CRing} , \textbf{Set}]$ is the Yoneda embedding $h^-$ applied to $R$), and whose morphisms $(R , \rho) \to (S , \sigma)$ are ring homomorphisms $\varphi: R \to S$ such that $\rho \circ h^\varphi = \sigma$. Define the diagram $D_\mathcal{F} : \textbf{P}_\mathcal{F}^\text{op} \to \textbf{LRS}$ to be the composition of the forgetful functor $\textbf{P}_\mathcal{F}^\text{op} \to \textbf{CRing}^\text{op}$ and the spec functor $\operatorname{Spec} : \textbf{CRing}^{\text{op}} \to \textbf{LRS}$. Explicitly it sends $(R , \rho) \mapsto \operatorname{Spec}(R)$.
In Demazure & Gabriel's Introduction to Algebraic Geometry and Algebraic Groups, the geometric realization of the functor $\mathcal{F}$ is defined by $| \mathcal{F} | := \operatorname{colim} D_\mathcal{F}$. In this book it is proved that the geometric realization functor $| – | : [\textbf{CRing} , \textbf{Set}] \to \textbf{LRS}$ is left adjoint to the functor $\textbf{LRS} \to [\textbf{CRing} , \textbf{Set}]$ which sends $X \mapsto \operatorname{Hom}_{\textbf{LRS}}(\operatorname{Spec}(-) , X)$.
In particular if $\mathcal{F}$ is representable by some scheme $Y$, i.e. $\mathcal{F} \cong \operatorname{Hom}_{\textbf{LRS}}(\operatorname{Spec}(-) , Y)$, then I think that $|\mathcal{F}| \cong Y$.
Question
My question is how does one go about computing the geometric realization of a functor which is not representable by schemes? For example what is the geometric realization of the functor $R \mapsto R^{\oplus \mathbb{N}}$? (This is basically the only non-representable functor I know.)
Small colimits in $\textbf{Set}$ are easy to construct (take the disjoint union and mod out by some equivalence relation), but the colimit in question here is certainly not small. Of course Demazure & Gabriel use Grothendieck universes throughout their book, so then $\textbf{P}_\mathcal{F}$ is a small category, however I'm not sure how to find a suitable Grothendieck universe to explicitly calculate this colimit for my example.
Best Answer
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:
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.