Inspired by this question.
Consider the set $[2n]=\{1,2,\dots,2n\}$ for some positive integer $n$. Let $A_1,\dots,A_k$ be subsets of $[2n]$ such that every subset of $[2n]$ is equal to $A_i\cup A_j$ for some $i,j$. What is the minimal $k$ possible?
A first bound is $k^2≥4^n$, so $k≥2^n$. Actually more since $k(k+1)/2\ge 4^n$.
Notice that combining all $a$-subsets with all $b$-subsets, where $b\ge a$, you obtain all the $c$-subsets where $b\le c \le a+b$, so an upper bound would be $k\le 1 + \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{7}+\dots$ choosing $A_i$ all subsets of cardinality $0,1,3,7,\dots,2^m-1$.