Photons propagating in Laguerre-Gaussian modes have characteristic orbital angular momenta, which are fundamental optical degrees of freedom. The orbital angular momentum of light has potential application in high-capacity optical communication and even in quantum information processing. In this work, we experimentally construct a ring cavity with four lenses and four mirrors that is completely degenerate for Laguerre-Gaussian modes. By measuring the transmitted peaks and patterns of different modes, the ring cavity is shown to support more than 31 Laguerre-Gaussian modes. The constructed degenerate cavity opens a new way for using the unlimited resource of available angular momentum states simultaneously.