Extensive numerical simulations are used to simulate the aggregation of colloidal particles conÿned in a two-dimensional space. The colloidal particles experience short-or medium-range repulsions coming from a potential barrier that inhibits the aggregation of the particles into the primary minimum of the potential. When the potential barrier is short ranged, we ÿnd the usual reaction-limited colloidal aggregation (RLCA) or di usion-limited colloidal aggregation cluster fractal dimensions for a high or low barrier, respectively. However, for medium-ranged, shallow barriers, for which the aggregation takes a long time as in the RLCA case, a very low fractal dimension is obtained, reaching values as low as 1.2 for the longer-ranged potentials considered. Nevertheless, as the aggregation proceeds, the cluster fractal dimension crosses over to a value close to the RLCA one, indicating that the small clusters of low fractal dimensionality act as the aggregating units of an RLCA system, given that they take a long time to react. A correspondence is made between our results and the experimental results by Hurd and Schaefer (Phys. Rev. Lett. 54 (1985) 1043).