High order corrections to the renormalon

High order corrections to the renormalon are considered. Each new type of insertion into the renormalon chain of graphs generates a correction to the asymptotics of perturbation theory of the order of ∼ 1. However, this series of corrections to the asymptotics is not the asymptotic one (i.e. the mth correction does not grow like m!). The summation of these corrections for the UV renormalon may change the asymptotics by a factor N^δ. For the traditional IR renormalon the mth correction diverges like (-2) ^m. However, this divergence has no infrared origin and may be removed by a proper redefinition of the IR renormalon. On the other hand, for IR renormalons in hadronic event shapes one should naturally expect these multiloop contributions to decrease like (-2) ^{- m}. Some problems expected upon reaching the best accuracy of perturbative QCD are also discussed.