(Inspired by [a meta thread on answers given in comments][1], I am recapping the answer given in the comments ([1](https://mathoverflow.net/questions/376289/there-are-at-most-four-mutually-visible-lattice-points#comment954042_376289) [2](https://mathoverflow.net/questions/376289/there-are-at-most-four-mutually-visible-lattice-points#comment954043_376289) [3](https://mathoverflow.net/questions/376289/there-are-at-most-four-mutually-visible-lattice-points#comment954044_376289)) 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.


  [1]: https://meta.mathoverflow.net/questions/4841/answers-in-comments-are-increasingly-more-common