We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent $2<γ<3$. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of $γ\approx 2$, clustering is virtually size independent and, at the same time, becomes a {\it de facto} non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes.