Twenty two years after the opening of the field of high temperature superconductivity by Georg Bednorz and Alex Müller a new addition to this class of materials, namely the iron-pnictides and chalcogenides, perplexes the scientific community. Making use of the accumulated wisdom, a great deal of their physics has been “mapped” in record time using all available measurement techniques. Among these, specific-heat has been proven a powerful and unique technique in the study of physical properties of bulk materials, especially of superconductors. We report here specific-heat measurements on the nearly optimally doped iconic iron-arsenide superconductor Ba0.59K0.41Fe2As2. We use a new method, based on direct comparisons of α-model expressions for the electron contribution with the measured total specific heat, to extract the electron contribution. It circumvents the need in the conventional analyses for an independent, necessarily approximate, determination of the lattice contribution, which is subtracted from the total to obtain the electron contribution, and it eliminates the consequent uncertainties in the electron contribution. For Ba0.59K0.41Fe2As2 the electron density of states is comprised of contributions from two electron bands with superconducting-state energy gaps differing by a factor 3.8, with 77% coming from the band with the larger gap. The vortex-state specific heat suggests nodeless gaps. Comparison of the normal-state density of states with band-structure calculations shows an extraordinarily large effective mass enhancement, for which there is no precedent in simple metals and no theoretical explanation.