How many chains are there in a finite power set?

Except for the degenerate $n=0$ case, the number of chains is $4$ times the $n$th Fubini number, A000670, also known as ordered Bell numbers.

The linked Wikipedia articles lists various ways formulas for these numbers, the most elementary ones being $$ a_n = \sum_{k=0}^n \sum_{j=0}^k (-1)^{k-j} \binom{k}{j}j^n \qquad\text{and}\qquad a_n \approx \frac{n!}{2(\log 2)^{n+1}}$$ (and then the number of chains is then $4 a_n$).

The factor of $4$ makes sense; it represents the fact that each of $\varnothing$ and $A$ can either be omitted or included in the chain without affecting its validity.