We present a study of the magnetospheric cusp response to extreme external parameters during passage of the ICME over the Earth on 10 November 2004, based on Cluster observations of the plasma properties inside the low-latitude boundary layer (LLBL)/cusp regions. Two separate events are observed while Cluster is in the dawn sector, 07-08 h magnetic local time (MLT). First, a LLBL/cusp crossing occurs during a period of strong southward IMF. During this time, the LLBL/cusp is very small, similar to 0.8-1 degrees invariant latitude (ILAT) and moves equatorward, down to 67 degrees ILAT. This can be explained by the occurrence of significant magnetopause erosion due to enhanced dayside sub-solar reconnection. The energy of the plasma inside this region is higher than normal, and the low-energy cut-off often observed in the ion data is also unusually high. This might be explained by the suggestion that the local magnetosheath Alfven velocity and deHoffmann-Teller velocity are also both extremely high. However, the plasma convection and parallel velocity inside this region are not very high. The second event discussed in this paper is a LLBL/cusp crossing during strong equatorial IMF (mostly due to the dominant dawn-dusk component). Under these conditions, occurring at the same time as pulses of solar wind dynamic pressure, the observations are very complicated. However, we suggest that in the polar region of the southern hemisphere, Cluster cross two LLBLs/cusps, spatially separated by polar cap plasma. The first LLBL/cusp is formed by anti-parallel reconnection in the dusk sector of the southern hemisphere and the second is formed by anti-parallel reconnection in the dawn sector of the northern hemisphere. The second LLBL/cusp is located at extremely low latitude, less than similar to 66.3 degrees ILAT. During all LLBL/cusp crossings, strong ionospheric O+ ion outflow is detected in the form of a narrow beam with limited pitch-angle range.