The blue compact dwarf galaxy NGC 5253 hosts a very young starburst containing twin nuclear star clusters, separated by a projected distance of 5 pc. One cluster (#5) coincides with the peak of the Hα emission and the other (#11) with a massive ultracompact H ii region. A recent analysis of these clusters shows that they have a photometric age of 1 ± 1 Myr, in apparent contradiction with the age of 3-5 Myr inferred from the presence of Wolf-Rayet features in the cluster #5 spectrum. We examine Hubble Space Telescope ultraviolet and Very Large Telescope optical spectroscopy of #5 and show that the stellar features arise from very massive stars (VMS), with masses greater than 100 M , at an age of 1-2 Myr. We further show that the very high ionizing flux from the nuclear clusters can only be explained if VMS are present. We investigate the origin of the observed nitrogen enrichment in the circum-cluster ionized gas and find that the excess N can be produced by massive rotating stars within the first 1 Myr. We find similarities between the NGC 5253 cluster spectrum and those of metal poor, high redshift galaxies. We discuss the presence of VMS in young, star-forming galaxies at high redshift; these should be detected in rest frame UV spectra to be obtained with the James Webb Space Telescope. We emphasize that population synthesis models with upper mass cut-offs greater than 100 M are crucial for future studies of young massive star clusters at all redshifts.