In this paper the authors study study the growth of the clusters for a class of aggregation models on the integer lattice \(\mathbb{Z} ^d, d \geqslant 2\), in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation. The authors observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the diffusion-limited aggregation model generalizes to a large class of such models. The authors use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Moreover the bound implies that the fractal dimension of ballistic aggregation clusters in \(\mathbb{Z} ^2\) is \(2\), and in this way the authors provide a proof for a long standing conjecture in the physics literature.