How many integer matrices with the spectral radius bounded by a fixed constant are there? Without any restrictions the answer is infinitely many. Indeed, $nJ_0$, where $J_0$ is the Jordan cell with the eigenvalue $0$ and $n$ is any integer, all have spectral radius $0$. On the other hand, for symmetric matrices the spectral radius is equal to the spectral norm so there are only finitely symmetric integer matrices with bounded spectral radius.
What about other conditions that rule out Jordan cells and other nilpotents? For example, matrices with strictly positive entries, invertible or diagonalizable matrices. I suspect that there are infinitely many invertible or diagonalizable ones, but cannot think of any general construction.
The motivation comes from trying to constructively generate large collections of invertible "random" integer matrices whose spectral radius stays between $1$ and $2$ (it cannot be less than $1$), so that their powers do not explode too quickly. This comes up in encryption.
EDIT: For integer matrices with strictly positive entries this is answered in Does small Perron-Frobenius eigenvalue imply small entries for integral matrices? on MathOverflow There are finitely many because the sum of all entries is bounded by the square of the spectral radius.