In [*Asymptotic Enumeration of Partial Orders on a Finite Set* (1975)][1], Kleitman and Rothschild showed that almost all partial orders on an $n$-element set have a simple description: they have three "layers" $L_1$, $L_2$, and $L_3$ of incomparable elements, of size $n/4$, $n/2$, and $n/4$ respectively.  Each element of $L_1$ is covered by about half the elements of $L_2$.  Likewise for $L_2$ and $L_3$, and in the reverse direction.  

So almost all finite posets have height $3$.  

(If you don't have the time/effort to read Kleitman and Rothschild's paper, I came across this reference via G. Brightwell, *Linear extensions of random orders*, Discrete Math. 125 (1994) pp.87-96.  If memory serves, Brightwell's paper presents a good summary of this one.) 

  [1]: https://www.jstor.org/stable/1997200