(Inspired by a meta thread on answers given in comments, I am recapping the answer given in the comments as a cw answer)

The largest number of mutually visible points in $\mathbb{Z}^d$ is $2^d$. This is achieved, for example, by the points $\lbrace 0, 1\rbrace^d$. Since there are $2^d$ orbits of $(2\mathbb{Z})^d$ in $\mathbb{Z}^d$ and two points in the same orbit are not mutually visble (the midpoint of the segment connecting them is a lattice point), no more than $2^d$ points can be pairwise mutually visible.

(Inspired by a meta thread on answers given in comments, I am recapping the answer given in the comments (1 2 3) as a CW answer.)

The largest number of mutually visible points in $\mathbb{Z}^d$ is $2^d$. This is achieved, for example, by the points $\lbrace 0, 1\rbrace^d$. Since there are $2^d$ orbits of $(2\mathbb{Z})^d$ in $\mathbb{Z}^d$ and two points in the same orbit are not mutually visble (the midpoint of the segment connecting them is a lattice point), no more than $2^d$ points can be pairwise mutually visible.