  We study the growth of fractal clusters in the Dielectric Breakdown Model
(DBM) by means of iterated conformal mappings. In particular we investigate the
fractal dimension and the maximal growth site (measured by the Hoelder exponent
$\alpha_{min}$) as a function of the growth exponent $\eta$ of the DBM model.
We do not find evidence for a phase transition from fractal to non-fractal
growth for a finite $\eta$-value. Simultaneously, we observe that the limit of
non-fractal growth ($D\to 1$) is consistent with $\alpha_{min} \to 1/2$.
Finally, using an optimization principle, we give a recipe on how to estimate
the effective value of $\eta$ from temporal growth data of fractal aggregates.
