Despite 35 years of investigation, much remains to be understood regarding charge transport in semiconducting polymers, including the ultimate limits on their conductivity. Recently developed ion gel gating techniques provide a unique opportunity to study such issues at very high charge carrier density. Here we have probed the benchmark polymer semiconductor poly(3-hexylthiophene) at electrochemically induced three-dimensional hole densities approaching 1021 cm-3 (up to 0.2 holes per monomer). Analysis of the hopping conduction reveals a remarkable phenomenon where wavefunction delocalization and Coulomb gap collapse are disrupted by doping-induced disorder, suppressing the insulator-metal transition, even at these extreme charge densities. Nevertheless, at the highest dopings, we observe, for the first time in a polymer transistor, a clear Hall effect with the expected field, temperature and gate voltage dependencies. The data indicate that at such mobilities (∼0.8cm2 V-1 s-1), despite the extensive disorder, these polymers lie close to a regime of truly diffusive band-like transport.