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(1). However, one major limitation has not been overcome; transport properties in polymer films are still limited by pervasive conformational and energetic disorder(2-5). This not only limits the rational design of materials with higher performance, but also prevents the study of physical phenomena associated with an extended pi-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(6), 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.