We investigate the pressure-temperature phase diagram of elemental potassium (K) up to multiterapascal (TPa) pressures using ab initio random structure searching (AIRSS), discovering eight structural phase transitions beyond the already known double hexagonal close-packed (dhcp) structure. Starting at 1.15 TPa, K transitions from the dhcp structure and passes through a variety of close-packed hexagonal and trigonal phases (dhcp $\ensuremath{\rightarrow}P{6}_{3}/mmc\ensuremath{\rightarrow}P3m1\ensuremath{\rightarrow}P\overline{6}m2)$, which differ only in their stacking sequence along the $c$ axis. At 2.55 TPa, K adopts the hexagonal close-packed (hcp) structure, and at 7.21 TPa transitions into a complex orthorhombic phase of $Fddd$ symmetry with 64 atoms in its (conventional) unit cell, before assuming an $Ibam$ structure at 8.60 TPa, and, eventually, the face centered cubic (fcc) structure at 23.6 TPa, which persists well into the petapascal (PPa) regime. Further, we calculate the full pressure-temperature dependence of the melting line of K through extensive molecular dynamics simulations. We study the evolution of the bonding topology of K with pressure, finding that K passes through two quasi-molecular phases featuring diatomic pairs (the $Fddd$ and $Ibam$ structures), before ultimately becoming a high pressure electride (HPE) in the fcc phase. The electron-phonon and superconducting properties of K at these extreme pressures are also investigated, where we find the critical temperature ${T}_{c}$ rises to a maximum of between 7.65 K and 15.70 K in the hcp phase, before eventually decreasing to essentially zero in the fcc phase. Our results fully elucidate the structural and electronic behavior of K under the most extreme conditions, and provide new case studies for multi-TPa dynamic compression experiments.