Successive interference cancellation (SIC), which allows one to recover additional information from otherwise lost collision signals, has been proposed in a variety of communication systems. In this paper, we introduce a random access protocol operating on channel supporting SIC that achieves a maximum stable throughput (MST) of 1. To the best of our knowledge, the highest MST achieved on such a channel thus far was 0.693 by the SICTA algorithm of Yu and Giannakis [1]. As opposed to the SICTA algorithm, the newly proposed algorithm has very limited practical value due to its unbounded computational complexity. That is, the computation time needed per time slot is finite, but unbounded over all time slots. To bound the computation time, we also propose a hybrid algorithm, which combines the new algorithm with SICTA. We demonstrate that the MST of this hybrid algorithm is 1 - ε, where ε > 0 can be chosen arbitrarily small given sufficient computational power.