There are two ways to cover a geometric shape with primitive units: you can allow the units to overlap, or require that they be disjoint. Of course the number of units in the case of disjoint covering may be larger. But how much larger?
If the units are squares, the number of units in a disjoint covering might be arbitrarily larger. For example, consider an $n\times (n+1)$ rectangle. For every $n$, it can be covered by two overlapping squares. But if the squares have to be disjoint, at least $O(\log n)$ squares are required.
What if the units are rectangles? Is there a sequence of shapes that can be covered by two rectangles, but whose disjoint-rectangles-cover-number can be arbitrarily large?
EDIT: This question is related: Tiling an L-shape with "almost square"s