Conjugated polymers enable the production of flexible semiconductor devices that can be processed from solution at low temperatures. Over the past 25 years, device performance has improved greatly as a wide variety of molecular structures have been studied. However, one major limitation has not been overcome; transport properties in polymer films are still limited by pervasive conformational and energetic disorder. This not only limits the rational design of materials with higher performance, but also prevents the study of physical phenomena associated with an extended π-electron delocalization along the polymer backbone. Here we report a comparative transport study of several high-mobility conjugated polymers by field-effect-modulated Seebeck, transistor and sub-bandgap optical absorption measurements. We show that in several of these polymers, most notably in a recently reported, indacenodithiophene-based donor-acceptor copolymer with a near-amorphous microstructure, the charge transport properties approach intrinsic disorder-free limits at which all molecular sites are thermally accessible. Molecular dynamics simulations identify the origin of this long sought-after regime as a planar, torsion-free backbone conformation that is surprisingly resilient to side-chain disorder. Our results provide molecular-design guidelines for 'disorder-free' conjugated polymers.