We investigate the behavior of the non-negative rank of a matrix, $\hbox{rk}_+$, on Kronecker product, $\times$. Beasley et al. [2013] constructed matrices with $\hbox{rk}_+(M\times M)$ strictly smaller than $\hbox{rk}_+(M)^2$.
We will show that i) there exists an $M$ with $\hbox{rk}_+(M\times M)\leq O(\hbox{rk}_+(M)\log(\hbox{rk}_+(M))$.

Defining asymptotic non-negative rank as $\overline{\hbox{rk}}_+(M):= \lim_{k\rightarrow \infty} \sqrt[k]{{\hbox{rk}}_+(M^{(k)})}$, where $M^{(k)}$ is the $k$-th Kronecker power, we show that
ii) there exists an $M$ with $\overline{\hbox{rk}}_+(M)\leq O(\log \hbox{rk}_+(M))$,
iii) there exist $M,N$ with $\overline{\hbox{rk}}_+(M\times N)<\overline{\hbox{rk}}_+(M)\overline{\hbox{rk}}_+(N)$.

We also refute a conjecture of Vandaele et al. [2016] that $\hbox{rk}_+(M\times M)\geq \Omega(\hbox{rk}_+(M)\hbox{rk}(M))$.