Three stages of dice whose counts come from previous sums

Roll one fair die and call the result $N$. Then roll $N$ fair dice and let $M$ be their total. Finally roll $M$ fair dice and let $S$ be their total. What is $\mathbb{E}[S]$?

$\tfrac{343}{8}$. For a random number $K$ of independent dice, the tower property gives $\mathbb{E}[\text{total}]=\tfrac72\,\mathbb{E}[K]$. Applying this twice, $\mathbb{E}[M]=(\tfrac72)^2$ and $\mathbb{E}[S]=(\tfrac72)^3=\tfrac{343}{8}$.