At right are solutions for the T- and Z-tetrominos.  Below is a solution for an octomino.

It's pretty obvious that any solution to this problem for n distinct polyominos has to cover 2ⁿ-1 squares, and the individual polyominos have to have 2^n-1 squares.  I will leave to someone else the discovery of a 16-omino, of which five copies can be combined to produce all 31 possible overlaps of one square each.  Let me know.