As society moves towards greater dependence on technology, understanding and predicting extreme Space Weather becomes increasingly important. Indeed, the complex, simultaneous interplay between multiple phenomena can unpack further insight into Space Weather effects. Thanks to decades of in-situ and ground observations, long archives of Space Weather measurements are ripe for exploitation by more complex, data intensive techniques. In this study, three decades of in-situ and ground observations of Earth’s Space environment are leveraged to model the joint extremal behaviour of solar wind driving and internal geomagnetic responses, using Bivariate Extreme Value Analysis. While often utilised in other fields with more long-term and/or higher spatial resolution datasets (such as meteorology), this is the first application of this technique in more than one dimension in the field of space plasma physics. An initial result is that upstream and internal variables exhibit weaker extremal dependence on minute timescales than on hourly timescales, in line with characteristic magnetospheric coupling timescales. Specifically, the joint extremal behaviour of a solar wind coupling function with the auroral electrojet index AE is explored and predicted. This modelling enables estimation of a 75-year return period for the 2003 Halloween geomagnetic storm.