[Partitioning a Rectangle...]

Start with a rectangle with at least two integer edges. (Just nitpicking since a rectangle can't have only one integer edge)

To create a larger rectangle by adding another rectangle with at least two integer edges there are two options.

Make them adjacent along the non integer edge which would add the two integer edge lengths to make the new rectangle's integer edge.

Make them adjacent along the integer edge which would make the new rectangle have an edge length equal to the integer edge of the two smaller rectangles.

Repeat. There's just no way to add the second rectangle without retaining at least two integer edges.

Any rectangle you add, if partitioning is possible, would be created the same way from the smallest rectangle.