In passive enhancement cavities the achievable power level is limited by mirror damage. Here, we address the design of robust optical resonators with large spot sizes on all mirrors, a measure that promises to mitigate this limitation by decreasing both the intensity and the thermal gradient on the mirror surfaces. We introduce a misalignment sensitivity metric to evaluate the robustness of resonator designs. We identify the standard bow-tie resonator operated close to the inner stability edge as the most robust large-mode cavity and implement this cavity with two spherical mirrors with 600 mm radius of curvature, two plane mirrors and a roundtrip length of 1.2 m, demonstrating a stable power enhancement of near-infrared laser light by a factor of 2000. Beam radii of 5.7 mm × 2.6 mm (sagittal × tangential 1/e² intensity radius) on all mirrors are obtained. We propose a simple all-reflective ellipticity compensation scheme. This will enable a significant increase of the attainable power and intensity levels in enhancement cavities.