I would like to collect arguments whose conclusion indicate that the dimension of space cannot be anything.
For example, the existence of non-trivial knots implies that the dimension of space is $3$ (since in dimension $1$, there are no knots, in dimension $2$, they are all circles by Jordan's theorem, and in dimension $4$, they are all isotopic to circles since there is enough room to unentangle any curve).
As another example, a radial, divergence-free vector field on $\mathbb{R}^n \setminus \{0\}$ is proportional to $x \mapsto \frac{x}{\vert x \vert^{n}}$; so if the gravitational field created by a point mass is physically supposed to be divergence-free and to have some form, then the dimension of space has to be $3$.
As a kind of a joke, a teacher once told me that in dimension $2$, living beings like us could not exist, since their digestive tubes would disconnect their bodies.
I know that some developments of string theory lead to results like this, but I would expect more elementary examples.
