Flexible electronic circuits are an essential prerequisite for the development of rollable displays, conformable sensors, biodegradable electronics and other applications with unconventional form factors. The smallest radius into which a circuit can be bent is typically several millimetres, limited by strain-induced damage to the active circuit elements. Bending-induced damage can be avoided by placing the circuit elements on rigid islands connected by stretchable wires, but the presence of rigid areas within the substrate plane limits the bending radius. Here we demonstrate organic transistors and complementary circuits that continue to operate without degradation while being folded into a radius of 100 μm. This enormous flexibility and bending stability is enabled by a very thin plastic substrate (12.5 μm), an atomically smooth planarization coating and a hybrid encapsulation stack that places the transistors in the neutral strain position. We demonstrate a potential application as a catheter with a sheet of transistors and sensors wrapped around it that enables the spatially resolved measurement of physical or chemical properties inside long, narrow tubes.