Most probable outcome of throwing a random number of dice. Say you have $2$ dice. The most probable outcome (sum of the values of the upper faces) would be $7$. But $2$ dice is a very arbitrary amount, so what happens if we use a dice to determine the amount of dice we throw?
$1)$ What would me the most probable outcome there?
And again, having only $1$ chooser die is very arbitrary and we could have a third "dice layer" (Right now we have $3$. The first layer are the dice that output the last number, the second layer is the dice that choose the amount of dice in the first layer and the third layer decides the number of dice in the second one), and so on.

Can we know what is the most probable outcome given n layers (Assuming that the last layer contains only $1$ die)?

 A: I don't know of a closed-form formula for this problem, but one can compute the distribution after $k$ rounds.  For $1$ round, each of the six values are equally probable.  If you know the probabilities of $1,2,\ldots,6 \cdot n$ when rolling up to $n$ dice, and you also know the probabilities of $m,m+1,\ldots,6 \cdot m$ when rolling $m$ dice for $1 \leq m \leq 6 \cdot n$, you can compute the probabilities of $1,2,\ldots,6 \cdot 6 \cdot n$ by applying the law of total probability.
A simple-minded program that does just that computes,
$$
\begin{array}{c|rrl}
\text{round} & \text{maximum} & \text{most probable} & \text{probability} \\\hline
2 & 36 & 6 & 0.0600387 \\
3 & 216 & 20 & 0.0145907 \\
4 & 1296 & 69 & 0.00411249 \\
5 & 7776 & 242 & 0.00117115 \\
6 & 46656 & 848 & 0.000334312
\end{array}
$$
The distribution for six rounds looks like this:

This is the program:

function multileveldice(rounds)
  % Compute maximum number of dice rolled at the last level.
  ndice = 6^(rounds-1);
  % Compute the distributions for rolling up to ndice dice.
  D = zeros(ndice,6*ndice);
  D(1,1:6) = 1/6;
  for n=2:ndice
    D(n,n:6*n) = conv(D(n-1,n-1:(n-1)*6),D(1,1:6));
  end
  % Compute the probability of each value.
  P = zeros(rounds,6*ndice);
  P(1,1:6) = 1/6;
  for n=2:rounds
    P(n,:) = P(n-1,1:ndice) * D;
  end

  % Print most likely value and its probability.  Plot distribution.
  [maxp, maxi] = max(P(rounds,:));
  fprintf('Maximum probability is %g for value %g\n', maxp, maxi)
  plot(1:ndice*6, P(rounds,:))
  grid on
end


