The adic spectrum $Spa(\mathbb{Z},\mathbb{Z})$ looks as follows:
First, there is a point for every prime ideal $\mathfrak{p}\subset\mathbb{Z}$, corresponding to the valuation given by the composition of the quotient map $\mathbb{Z}\rightarrow \mathbb{Z}/\mathfrak{p}$ with the trivial valuation. Then there is an additional point for every $p$-adic valuation.
The points corresponding to maximal ideals are closed, a point corresponding to the $p$-adic valuation has the point corresponding to $(p)$ in its closure and the point corresponding to the zero ideal is a generic point.
It is known that this space is homeomorphic to the spectrum of a ring by a general result by M. Hoechster, found in "Prime Ideal Structure in Commutative Rings". However I am asking just out of curiosity and would like to avoid to get into the technical details of the construction and wondered if anybody has figured out this special case.