The growth process of higher fullerenes through adduction of small carbon clusters and carbon atoms is studied using a molecular-dynamics simulation method. Small clusters, such as C3 and C2, and carbon atoms easily adsorb on the surface of fullerene cages, which have coordination-number defects or topological defects, when they collide at thermal velocities with the fullerenes. Annealing the chemisorption complexes at 3000K, the attached clusters are soon incorporated into the network of the fullerene cages, via a self-assembly growth process. Although the fullerenes grown in this way usually have defects, their ring structures can be described by a formula deduced from Euler's theorem. The energies of the fullerenes remarkably decrease with their increasing size, when they are annealed to their low-lying energy structures.