Can expected "depth" of an element and expected "height" differ significantly?
In chained hash tables with uniform hashing a similar question arises. Given a table with $n$ elements hashed into $m$ buckets, the expected number of elements per bucket is $n/m$, but what is the expected length of the longest chain?

I found a Stackoverflow question about chained hash tables. The answer by btilly argues that for fixed ratio $n/m$ the expected worst case longest chain is $\Theta(\log n/ \log\log n)$.  (Since $n/m$ is fixed, $m$ doesn't need to come into the answer.)

And the answer by templatetypedef references the following paper for a more detailed analysis:
Gaston H. Gonnet: Expected Length of the Longest Probe Sequence in Hash Code Searching.
J. ACM_ 28(2):289-304, 1981.
[Free techreport version at University of Waterloo]

Slightly more generally I found a math.stackexchange question about the balls and bins problem.  The answer by Yuval Filmus cites the following paper, which gives even tighter bounds:
Martin Raab; Angelika Steger: Balls into Bins - A Simple and Tight Analysis. 2nd Intl Wkshp on Randomization and Approximation Techniques in Computer Science, pp. 159-170, 1998.

Finally, there is apparently a branch of statistics called extreme value theory culminating in the Fisher-Tippett-Gnedenko theorem, which, I think, gives expected maxima for a huge variety of distributions.