Extreme astrophysical events such as relativistic flows, cataclysmic explosions and black hole accretion are one of the key areas for astrophysics in the 21st century. The extremes of physics experienced in these environments are beyond anything achievable in any laboratory on Earth, and provide a unique glimpse at the laws of physics operating in extraordinary regimes. All of these events are associated with transient radio emission, a tracer both of the acceleration of particles to relativistic energies, and coherent emitting regions with huge effective temperatures. By studying radio bursts from these phenomena we can pinpoint the sources of explosive events, understand the budget of kinetic feedback by explosive events in the ambient medium, and probe the physical state of the universe back to the epoch of reionisation, less than a billion years after the big bang. In seeking to push back the frontiers of extreme astrophysics, I will use a trio of revolutionary new radio telescopes, LOFAR, ASKAP and MeerKAT, pathfinders for the Square Kilometre Array, and all facilities in which I have a major role in the search for transients. I will build an infrastructure which transforms their combined operations for the discovery, classification and reporting of transient astrophysical events, over the whole sky, making them much more than the sum of their parts. This will include development of environments for the coordinated handling of extreme astrophysical events, in real time, via automated systems, as well as novel techniques for the detection of these events in a sea of noise. I will furthermore augment this program by buying in as a major partner to a rapid-response robotic optical telescope, and by cementing my relationship with an orbiting X-ray facility. This multiwavelength dimension will secure the astrophysical interpretation of our observational results and help to revolutionise high-energy astrophysics via a strong scientific exploitation program.