In this work, a growing network model that can generate a random network with finite degree in infinite time is studied. The dynamics are governed by a rule where the degree increases under a scheme similar to the Malthus-Verhulst model in the context of population growth. The degree distribution is analysed in both stationary and time-dependent regimes through some exact results and simulations, and a scaling behaviour is found in asymptotically large time. For finite times, the time-dependent degree distribution displays an accumulation of hubs as a result of competition between attractive and repulsive terms in linking probability.