The Karpelevič theorem will make it difficult for the matrix not to at least involve a permutation matrix (notice that the upper-left $2$-by-$2$ block of the matrix given by Gerry is a permutation matrix).

If $C_n$ is the $n$-by-$n$ defined by $C_n = [e_n \mid e_1 \mid \cdots \mid e_{n-1}]$, i.e., $C_n$ is the adjancency matrix of a directed $n$-cycle, where $e_i$ denotes the $i^\text{th}$ canonical basis vector of $\mathbb{C}^n$, then the $n$-by-$n$ (row) stochastic matrix
$$ M_n :=
\begin{bmatrix}
0 & e_1^\top \\
0 & C_{n-1}
\end{bmatrix}$$
has spectrum $\sigma(M) = \{ 0, 1, \omega,\dots,\omega^{n-1}\}$, where $\omega:=\exp(2\pi i/n)$. Thus, $M_n$ has $n-2$ eigenvalues (distinct from unity) on the unit-circle.

For instance, the matrix
$$ M_5=
\begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 \\
\end{bmatrix} $$
has spectrum $\{0,\pm 1, \pm i\}$.