We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for such ODEs requires a computational cost of order $\varepsilon^{-3}$ to achieve a root-mean-square error of size $\varepsilon$. In this work we adapt recently introduced full history recursive multilevel Picard (MLP) algorithms to reduce this computational complexity. Our main result shows for every $δ>0$ that the proposed MLP approximation algorithm requires only a computational effort of order $\varepsilon^{-(2+δ)}$ to achieve a root-mean-square error of size $\varepsilon$.