  We establish some scaling limits for a model of planar aggregation. The model
is described by the composition of a sequence of independent and identically
distributed random conformal maps, each corresponding to the addition of one
particle. We study the limit of small particle size and rapid aggregation. The
process of growing clusters converges, in the sense of Caratheodory, to an
inflating disc. A more refined analysis reveals, within the cluster, a tree
structure of branching fingers, whose radial component increases
deterministically with time. The arguments of any finite sample of fingers,
tracked inwards, perform coalescing Brownian motions. The arguments of any
finite sample of gaps between the fingers, tracked outwards, also perform
coalescing Brownian motions. These properties are closely related to the
evolution of harmonic measure on the boundary of the cluster, which is shown to
converge to the Brownian web.
