The number of times that max is assigned to is known as the number of records (or left-to-right maxima) in the permutation. The following results are standard, and can be found in a paper of Kortchemski, which gives a much more refined analysis.
Lemma 1.
Let $X_1,\ldots,X_n$ be indicator variables for the events that the $i$th element of a random permutation is a record (larger than all previous elements). Then $\Pr[X_i=1] = 1/i$ and the $X_i$ are independent.
Corollary.
The expected number of records is $H_n = \log n + \gamma + O(1/n)$ (the $n$th harmonic number).
This follows from linearity of expectation.
Lemma 2.
The generating function for the number of records is
$$ T_n(q) = q(q+1)\cdots(q+n-1). $$
The coefficient $c(n,k)$ of $q^k$, which equals the number of permutations in $S_n$ having $k$ records, is the $(n,k)$ Stirling number of the first kind.
Corollary. The variance of the number of records is $\log n + \gamma - \pi^2/6 + O(1/n)$.
Indeed, notice that $T''_n(1) = \sum_k k(k-1) c(n,k)$. On the other hand,
$$
T''_n(q) = \sum_{0 \leq i \neq j \leq n-1} \frac{T_n(q)}{(q+i)(q+j)},
$$
and substituting $q = 1$,
$$
\frac{T''_n(1)}{n!} = \sum_{1 \leq i \neq j \leq n} \frac{1}{ij} = H_n^2 - \sum_{i=1}^n \frac{1}{i^2}.
$$
Denoting by $R$ the number of records, this is a formula for $\mathbb{E}[R(R-1)]$. Now
$$
\mathbb{V}[R] = \mathbb{E}[R(R-1)] + \mathbb{E}[R] - \mathbb{E}^2[R] =
H_n^2 - \sum_{i=1}^n \frac{1}{i^2} + H_n - H_n^2 = H_n - \sum_{i=1}^n \frac{1}{i^2}.
$$
It is well-known that $\sum_{i=1}^\infty 1/i^2 = \pi^2/6$, and we can estimate $\sum_{i=n+1}^\infty \leq \int_n^\infty dx/x^2 = 1/n$. In particular,
$$
\mathbb{V}[R] = H_n - \frac{\pi^2}{6} + O(1/n).
$$