We generate the sequence $\{a\}$ in the following way. $n$ is a given fixed integer, and for each position $i, a_i$ will be chosen from $\{1,2,...n\}$ with equal probability and the sequence ends after the first occurrence of the number $n$. Note that the length of the sequence isn't necessarily $n$, but we can figure out that the expected length of the sequence is $n$. What's the expected number of the maximum occurrence in such a sequence? Here the maximum occurrence means the number of occurrences of the most frequent number in the sequence. An asymptotic when $n$ tends to be infinity is also acceptable. Thanks a lot!
Additional Materials that may help: The expectation of the length is $n$. And for such a sequence with length $n$, there exists some paper proving the answer is in $\Theta(\frac{logn}{loglogn})$ scale, but I'm not quite sure if the conclusion still holds under my assumption.
Here's the link to the paper: https://www.ic.unicamp.br/~celio/peer2peer/math/balls-into-bins.pdf