This is somewhat inspired by Factoring a function from a finite set to itself.

Fix natural number $n$ and let $[n] := \{1,2,\ldots,n\}$. Set $g_0 \colon [n]\to [n]$ to be the identity, and for $i \geq 1$ define $g_i := f_i \circ g_{i-1}$ where the $f_i\colon [n] \to [n]$ are chosen (independently and) uniformly at random among all functions from $[n]$ to $[n]$.

What is the expected value of the smallest $t$ for which $g_t$ is a constant function? (More generally, what is the distribution of this random variable $t$?)

EDIT: As Peter Taylor explained, it is easy to view this also as a Markov chain on $[n]$ where at time $t$ our state $a_t$ is the size of the image of $g_t$. And as I mentioned in the comments then the trajectory of this Markov chain $(a_1-1,a_2-1,\ldots)$ gives a random partition with part sizes $\leq n-1$; the expected time to a constant function is the expected length of this partition.

There is also a natural $q$-analog of this problem, where instead of random functions $[n]\to [n]$ we look at random linear functions $\mathbb{F}_q^n\to \mathbb{F}_q^n$. This gives a Markov chain on $\{0,1,\ldots,n\}$ where our state $a_t$ is the dimension of the image of $g_t$. Of course now the transition probabilities of the Markov chain involve the parameter $q$ (and should recover the previous case with $q=1$).