Let $A$ be a finite set with $n$ elements. How many chains are there in $\mathcal P(A)$ -- that is, how many different subsets of $\mathcal P(A)$ are totally ordered by inclusion?
It's easy enough to count maximal chains; there are $n!$ of them. But counting all chains gets me into horrible inclusion-exclusion situations even if I try to do it by hand for small $n$.
Of course this is also the number of chains in a Boolean algebra with $2^n$ elements, or the number of chains in a finite lattice with $2^n$ elements.
By hand computation, the number of chains for $n$ running from $0$ to $6$ is $2, 4, 12, 52, 300, 2164, 18732$. This sequence appears to be unknown in OEIS.