  In this paper we consider weakly holomorphic modular forms (i.e. those
meromorphic modular forms for which poles only possibly occur at the cusps) of
weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a
distinguished set of generators $f_{m,2-k}$. Such weakly holomorphic modular
forms have been classified in terms of finitely many Eisenstein series, the
unique weight 12 newform $\Delta$, and certain Faber polynomials in the modular
invariant $j(z)$, the Hauptmodul for $\SL_2(\Z)$. We employ the theory of
harmonic weak Maass forms and (non-holomorphic) Maass-Poincar\'e series in
order to obtain the asymptotic growth of the coefficients of these Faber
polynomials. Along the way, we obtain an asymptotic formula for the partial
derivatives of the Maass-Poincar\'e series with respect to $y$ as well as
extending an asymptotic for the growth of the $\ell$-th repeated integral of
the Gauss error function at $x$ to include $\ell\in \R$ and a wider range of
$x$.
