Direction-of-arrival (DoA) estimation is a mature topic with decades of history. Despite the progress in the field, very few papers have looked at the problem of DoA estimation with unknown dynamic range. Consider the case of space exploration or near-field and far-field emitters. In such settings, the amplitude of the impinging wavefront can be much higher than the maximum recordable range of the sensor, resulting in information loss via clipping or sensor saturation. In this paper, we present a novel sensing approach for DoA estimation that exploits hardware-software co-design and is pivoted around the theme of unlimited sensing. On the hardware front, we capitalize on a radically new breed of analog-to-digital converters (ADCs) which, instead of saturating, produce modulo measurements. On the algorithmic front, we develop a mathematically guaranteed DoA estimation technique which is non-iterative and backwards compatible with existing DoA algorithms. Our computer experiments show the efficiency of our approach by estimating DoAs from signals which are orders of magnitude higher than the ADC threshold. Hence, our work paves a new path for inverse problems linked with DoA estimation and at the same time provides guidelines for new hardware development.