Large solar energetic particle (SEP) events constitute a serious radiation hazard to astronauts and spacecraft systems. It is essential to determine the highest particle intensities reached in SEP events, especially at the energies that pose serious risks to human health and spacecraft performance. It has been argued that the highest particle intensities measured during large SEP events occur in association with the passage of shocks driven by coronal mass ejections known as the energetic storm particle (ESP) component. Furthermore, it has been argued that the intensities measured early in the SEP events (known as the prompt component) are bounded by a maximum intensity plateau that results from wave-particle interactions that restrict the free streaming of particles (also called the “streaming limit”). We analyze proton intensities measured by the GOES spacecraft at the energy channels P5 (∼39–82 MeV) and P7 (∼110–500 MeV) during solar cycles 22 and 23 and examine whether the highest intensities were measured during the prompt or the ESP components of the SEP events. We find three (one) SEP events in which the highest proton intensities measured during the prompt component at the energy channel P5 (P7) exceeded by a factor of 4 or more the previously determined “streaming limit”. Arguments to explain intensities during the prompt components exceeding this limit invoke interplanetary conditions that inhibit the amplification of waves resonating with the streaming particles and/or the presence of interplanetary structures able to confine and/or mirror energetic particles. We analyze these possibilities for each one of these events.