This is just a speculative question I've been wondering about; I hope that others may find it interesting too, but if it's too vague please let me know!
We can picture $\mathbb{A}^1_{\mathbb{Z}} = \text{Spec}\mathbb{Z}[x]$ as a funny surface like the one below taken from Mumford's Red Book
with "vertical" lines corresponding to all the primes, and various horizontal lines, curves etc intersecting these corresponding to different algebraic numbers, or equivalently different nonmaximal prime ideals of $\mathbb{Z}[x]$.
We can also construct $\mathbb{P}^1_{\mathbb{Z}} = \text{Proj}\mathbb{Z}[x]$ which splits up as a disjoint union $\mathbb{P}^1_\mathbb{Z} \cong \mathbb{A}^1_{\mathbb{Z}} \sqcup \text{Spec}\mathbb{Z}$ consisting of the open subset isomorphic to the affine space $\mathbb{A}^1_\mathbb{Z}$ plus a closed "line at infinity" isomorphic to $\text{Spec}\mathbb{Z}$. The natural place for one to place this line at infinity in the picture above (as it is done in Eisenbud and Harris' The Geometry of Schemes) is as a horizontal line right at the top of the picture "above" all the other horizontal lines. It's a vague picture like many others associated to schemes over $\mathbb{Z}$, but it makes some degree of sense.
Granted this, we managed to "compactify" the vertical axis of $\mathbb{A}^1_\mathbb{Z}$ by adding the horizontal line at infinity isomorphic to $\text{Spec}\mathbb{Z}$. Is there a similar thing we can do for the horizontal axis? Given the infinitude of the primes, we'd need to have something like a fibre over an "infinite prime", possibly made rigorous using valuation theory. (From a geometrical viewpoint I wouldn't say the generic fibre of the morphism $\mathbb{A}^1_{\mathbb{Z}}\to\text{Spec}\mathbb{Z}$ was this new vertical line because I consider this fibre to be "spread out" over the whole of $\mathbb{A}^1_{\mathbb{Z}}$, but again this is just because of how I think about generic points - perhaps somebody can correct me on this.)
I don't think this new $\mathbb{A}^1_{\mathbb{Z}}$ with a "compactified arithmetic axis" would be a scheme, but is there a way we can enlarge the category of schemes to formalise this compactification for "infinite primes", e.g. in the sense of valuations?