Kissing number in 8 dimensions is supposedly 240, for spheres that are centered around points of E8 lattice.
However, if we simply take an 8-cube, it has 256 vertices. Using each vertex as a center of a sphere, we can put another sphere in the middle of this cube, touching all of them. This alone is already more than 240.
In addition, this middle sphere would actually touch the centers of all 16 7-faces (because the diagonal of unit 8-cube is exactly 2, and thus the middle sphere has a radius of 1/2, touching all of 7-faces), and thus 16 additional spheres in the neighboring cubes, giving total kissing number of 272.
Why doesn't this argument work?
The diagonal of an 8-dimensional cube of length 2 is $4 \sqrt{2}$. You can fit 256 unit spheres on the corners. However, the unit sphere in the center will not touch any corner sphere since $1+2+1 < 4 \sqrt{2}$.