We investigate the kinetic behaviour of the growth of aggregates through monomer birth and death and propose a simple model with the rate keels K(k) ∝ ku and K′(k) ∝ kv at which the aggregate Ak of size k respectively yields and loses a monomer. For the symmetrical system with K(k) = K′(k), the aggregate size distribution approaches the conventional scaling form in the case of u ＜ 2, while the system may undergo a gelation-like transition in the u ＞ 2 case. Moreover, the typical aggregate size S(t) grows as t1/(2-u) in the u ＜ 2 case and increases exponentially with time in the u = 2 case. We also investigate several solvable systems with asymmetrical rate keels and find that the scaling of the aggregate size distribution may break down in most cases.