In this study, the maximum spherical inversion in the maximum space, the Cartesian 3-space endowed with the maximum metric is defined and introduced. Firstly, the formula for computing the inverse of a point P with respect to the maximum sphere is given. Then, some basic properties of the maximum spherical inversion map are studied. The obtained results related to invariant line, the plane and the maximum sphere under the maximum spherical inversion are presented. In addition, the cross ratio and the harmonic conjugate in maximum space are given. Afterwars, the maximum spherical inverses of the cross ratio and the harmonic conjugate are examined.