  We study a random aggregation process involving rectangular clusters. In each
aggregation event, two rectangles are chosen at random and if they have a
compatible side, either vertical or horizontal, they merge along that side to
form a larger rectangle. Starting with $N$ identical squares, this elementary
event is repeated until the system reaches a jammed state where each rectangle
has two unique sides. The average number of frozen rectangles scales as
$N^\alpha$ in the large-$N$ limit. The growth exponent $\alpha=0.229\pm 0.002$
characterizes statistical properties of the jammed state and the time-dependent
evolution. We also study an aggregation process where rectangles are embedded
in a plane and interact only with nearest neighbors. In the jammed state,
neighboring rectangles are incompatible, and these frozen rectangles form a
tiling of the two-dimensional domain. In this case, the final number of
rectangles scales linearly with system size.
