What a silly problem.We can perform at most $n(n-1)/2$ moves.Consider the number of pairs of sets $A,B$ such that $A$ is a subset of $B$.It is easy to prove that this number is strictly increasing by at least $1$ per move,so this number is at least $0$ and at most $n(n-1)/2$ so we can perform at most $n(n-1)/2$ moves.Now,let's show an example.Consider intervals $[0,2^n],[2^n,2*2^n],..[2^n*(n-1),2^n*n]$.A1 will contain all elements from the first interval,A2 will contain the first 2^(n-1) elements from the first and the second interval,...$Ai$ will contain the first 2^(n-i+1) elements from the first $i$ intervals.Now,perform moves as follows(when we perform (Ai,Aj) $i<j$ then Aj becomes AiUAj):(A2,A1),(A3,A1),(A3,A2),...(Ai,A1),...(An,A(n-1)).