In this paper, we propose a hypernetwork model with a nonlinear preferential attachment, and study the evolving mechanism and topological properties of the hypernetwork. We analyze the model by using a Poisson process theory and a continuous technique, and give a characteristic equation of hyperdegrees. We obtain the stationary average hyperdegree distribution of the hypernetwork by the characteristic equation. The analytical result shows that the hypernetwork has a phenomenon of the rich get richer, and it accords well with the simulation. It is shown in this paper that the hyperdegree distribution of the dynamic model exhibits a stretched exponential distribution with the increase of the hypernetwork size. It proves that the rich get richer does not necessarily induce a power-law distribution.