In general, a completion of a normed field is naturally a colimit, since you can make a poset of normed fields generated by equivalence classes of sets of Cauchy sequences. I don't think there is an inverse limit construction of the reals, but it may be interesting to analyze the failure modes of some attempts.

First, we should take a closer look at the p-adic construction. We start by restricting our view to a valuation ring, namely the subring of elements of absolute value at most one. Then, we take an inverse limit over quotients by powers of the maximal ideal. Finally, we take the fraction field of the resulting complete ring.

The first step in the construction is already problematic with the archimedean valuation, because it is not an ultrametric, i.e., the set of rationals of norm at most 1 is not closed under addition. We can boldly press on, noting that we still have a continuous multiplicative monoid structure, and it has a maximal ideal $M = \{ x \in \mathbb{Q} : |x| < 1 \}$. Unfortunately, positive powers of M are equal to M itself, so the inverse limit of the quotients is just the group $\pm 1$, which is not the closed unit disk. We could also try the inverse limit of quotients by the directed system of ideals $I_\alpha = \{x \in \mathbb{Q} : |x| < \alpha \}$, but all of the quotients are totally disconnected. This is bad, because (if I'm not mistaken) the inverse limit of a directed system of totally disconnected spaces is totally disconnected, while the interval $[-1,1]$ (often called the "ring of integers of $\mathbb{R}$") is what we really want, and its connected component(s) are far from singletons.

In general, a completion of a normed field is naturally a colimit, since you can make a poset of normed fields generated by equivalence classes of sets of Cauchy sequences. I don't think there is an inverse limit construction of the reals, but it may be interesting to analyze the failure modes of some attempts.

First, we should take a closer look at the p-adic construction. We start by restricting our view to a valuation ring, namely the subring of elements of absolute value at most one. Then, we take an inverse limit over quotients by powers of the maximal ideal. Finally, we take the fraction field of the resulting complete ring.

The first step in the construction is already problematic with the archimedean valuation, because it is not an ultrametric, i.e., the set of rationals of norm at most 1 is not closed under addition. We can boldly press on, noting that we still have a continuous multiplicative monoid structure, and it has a maximal ideal $M = \{ x \in \mathbb{Q} : |x| < 1 \}$. Unfortunately, positive powers of M are equal to M itself, so the inverse limit of the quotients is just the monoid $\{-1,0,1\}$, which is not the closed unit disk. We could also try the inverse limit of quotients by the directed system of ideals $I_\alpha = \{x \in \mathbb{Q} : |x| < \alpha \}$, but all of the quotients are totally disconnected. This is bad, because (if I'm not mistaken) the inverse limit of a directed system of totally disconnected spaces is totally disconnected, while the interval $[-1,1]$ (often called the "ring of integers of $\mathbb{R}$") is what we really want, and its connected component(s) are far from singletons.