The object of this challenge is to tile rectangles and squares with squares in a "nowhere-neat" way. Nowhere-neat means that no two squares share two vertices and an entire edge.
A secondary challenge is to tile rectangles and squares without same-size squares sharing any segment of edge. These may be called "no-touch" solutions.
Tilings on this page: 11x15, 16x16, 19x19, 20x20, 20x85, 22x40, 23x23 (*), 27x27 (*), 28x43, 28x48, 32x32 (*, two solutions), 35x35 (*, two solutions), 37x37 (*), 38x38 (*), 39x39, (two solutions), 40x40 (two solutions, one *), 45x45, 47x47, 53x53 (two solutions), 55x55, 56x56, 59x59 (two solutions), 60x60, 65x65, 68x68, 71x71, 75x75, 83x83, 98x98, 105x105
(*) Different than the solution(s) on Karl's page for this size.