  In analogy to recent results on non-universal roughening in surface growth
[Lam and Sander, Phys. Rev. Lett. {\bf 69}, 3338 (1992)], we propose a variant
of diffusion-limited aggregation ($DLA$) in which the radii of the particles
are chosen from a power law distribution. For very broad distributions, the
huge particles dominate and the fractal dimension is calculated exactly using a
scaling theory. For narrower distributions, it crosses back to DLA. We
simulated $1200$ clusters containing up to $200,000$ particles. The fractal
dimensions obtained are in reasonable agreement with our theory.
