$\newcommand{\Si}{\Sigma}$Let $X_n$ be the maximum number of prank cigarettes any pack receives, so that $M_n=EX_n$. Note that $X_n=\max(N_1,\dots,N_n)$, where $(N_1,\dots,N_n)$ has the multinomial distribution with parameters $n;\frac1n,\dots,\frac1n$. So, by Theorem 1 or formula (6) of Raab and Steger,
\begin{equation*}
X_n\sim r_n:=\frac{\ln n}{\ln\ln n} \tag{1}\label{1}
\end{equation*}
in probability (as $n\to\infty$).
We will also prove
Proposition 1:
\begin{equation*}
EX_n^2\ll r_n^2.
\end{equation*}
(As usual, we write $A\ll B$ to mean $A=O(B)$.)
By Proposition 1 and the de la Vallée-Poussin theorem, $X_n/r_n$ is uniformly integrable, which implies
\begin{equation*}
M_n=EX_n\sim r_n
\end{equation*}
(which agrees with Qiaochu Yuan's heuristics/conjecture).
Proof of Proposition 1:
Let $[n]:=\{1,\dots,n\}$. Note that
\begin{equation*}
EX_n^2\ll\sum_{m\in[n]}mP(X_n\ge m)
=\Si_1+\Si_2, \tag{3}\label{3}
\end{equation*}
where
\begin{equation*}
\Si_1:=\sum_{1\le m\le 4r_n}mP(X_n\ge m),\quad \Si_2:=\sum_{4r_n<m\le n}mP(X_n\ge m).
\tag{4}\label{4}
\end{equation*}
It is easy to bound $\Si_1$:
\begin{equation}
\Si_1\le\sum_{1\le m\le 4r_n}m\ll r_n^2. \tag{5}\label{5}
\end{equation}
Let us now bound $\Si_2$.
Recall that $X_n=\max(N_1,\dots,N_n)$, where $(N_1,\dots,N_n)$ has the multinomial distribution with parameters $n;\frac1n,\dots,\frac1n$. Note that for each $m\in[n]$
\begin{equation*}
P(X_n\ge m)\le\sum_{i\in[n]}P(N_i\ge m)
=nP(N_1\ge m). \tag{7}\label{7}
\end{equation*}
Next, $N_1$ has the binomial distribution with parameters $n$ and $\frac1n$, and hence
\begin{equation*}
\begin{aligned}
P(N_1\ge m)&=\sum_{k=m}^n\binom nk\frac1{n^k}\Big(1-\frac1n\Big)^{n-k} \\
&\le\sum_{k=m}^n\binom nk\frac1{n^k}\le\sum_{k=m}^n\frac1{k!}\ll\frac1{m!}.
\end{aligned}
\tag{11}\label{11}
\end{equation*}
Further, for $m>4r_n$, eventually (that is, for all large enough $n$),
\begin{equation*}
m!\ge(m/e)^m=\exp(m\ln(m/e)) \\
\ge\exp\Big(4\frac{\ln n}{\ln\ln n}\,\ln\frac{\ln n}{\ln\ln n}\Big)\ge n^3. \tag{13}\label{13}
\end{equation*}
So, by \eqref{4}, \eqref{7}, \eqref{11}, and \eqref{13},
\begin{equation*}
\Si_2\le\sum_{4r_n<m\le n}mnP(N_1\ge m) \\
\ll\sum_{4r_n<m\le n}mn\frac1{n^3}\le1\ll r_n^2. \tag{15}\label{15}
\end{equation*}
Now Proposition 1 follows immediately from \eqref{3}, \eqref{5}, and \eqref{15}. $\quad\Box$
Quite similarly one can show that
\begin{equation*}
EX_n^p\sim r_n^p
\end{equation*}
for each real $p>0$.
