We study the maximal degree of (sub)critical Lévy trees which arise as the scaling limits of Bienaymé-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the Lévy tree conditioned to have a fixed maximal degree. In the case where the Lévy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned Lévy tree can be obtained by grafting a Lévy forest on an independent size-biased Lévy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the Lévy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.