  Let $\Phi = \{\phi_e\}_{e\in E}$ be a finitely irreducible conformal graph
directed Markov system (CGDMS) with symbolic representation $E_A^{\infty}$ and
limit set $J$. Under a mild condition on the system, we give a multifractal
analysis of level sets of Birkhoff averages with respect to Hausdorff dimension
for a large family of functions. We then apply these results to a few examples
in the case of both $E$ finite and $E$ countably infinite.
