The question is ill-posed, because you can always make a small deformation of a lattice to send the kissing number to zero. Even if you expand the spheres as much as possible, the kissing number will generically be 2. Even if you require linearly independent kissing points, then I think that generically there will only be 2n of them.

You could look at unimodular lattices. By definition, this is a lattice such that if you write the Euclidean dot product in a basis of the lattice, its matrix has integer entries and determinant 1. If you drop either determinant 1 or integer entries, you're back to the deformation instability. There are only finitely many unimodular lattices in any fixed dimension, and not very many in low dimensions. I don't know the behavior of unimodular kissing numbers in high dimensions.

The question is seems ill-posed, because you can always make a small deformation of a lattice to send the kissing number to zero. Even if you expand the spheres as much as possible, the kissing number will generically be 2. Even if you require linearly independent kissing points, then I think that generically there will only be 2n of them.

You could look at unimodular lattices. By definition, this is a lattice such that if you write the Euclidean dot product in a basis of the lattice, its matrix has integer entries and determinant 1. If you drop either determinant 1 or integer entries, you're back to the deformation instability. There are only finitely many unimodular lattices in any fixed dimension, and not very many in low dimensions. I don't know the behavior of unimodular kissing numbers in high dimensions.

Gil's answer is better. My comments make sense in low dimensions. But in high dimensions, people have a lot of trouble making lattices with large kissing numbers at all. That turn of events didn't occur to me.