\begin{aligned}
 \prod_{j\in\omega}(\omega+(\omega+1)^j)
 & = \sup_{n\in\omega}\prod_{j\in n}(\omega+(\omega+1)^j) \\
 & = \sup_{n\in\omega}\big((\omega+1)\cdot(\omega\cdot 2+1)\cdot(\omega^2+\omega+1)\cdot\cdots\cdot(\omega^{n-1}+\omega^{n-2}+\cdots+\omega+1)\big) \\
 & = \sup_{n\in\omega} \big(\omega^{n(n-1)/2+1}+\text{ne\v{s}to}\big) \\
 & = \omega^\omega
\end{aligned}