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Super-Eddington accretion periods are expected to be extremely important to explain the fast growth of the first supermassive black holes. We propose here a systematic broad-band X-ray study of super-Eddington AGN openPar Edd = 15 to Edd = 427closePar at z < 0.1 by simultaneously observing five of them with XMM-Newton and NuSTAR. We use the best sample available comma SEAMBHs comma which contain objects with black hole masses estimated from reverberation mapping comma making the Eddington ratio estimates extremely reliable. The broad-band observations will allow to shed light on a large number of fundamental issues comma such as the behaviour of the cutoff energy comma of the reprocessed radiation comma of the iron complex and of the soft excess at extreme Eddington ratios.