This paper proposes an adaptive transmission algorithm for slotted random access systems supporting the successive interference cancellation (SIC) at the access point (AP). When multiple users transmit packets simultaneously in a slot, owing to the SIC technique, the AP is able to decode them through SIC resolve procedures (SRPs), which may occupy multiple consequent slots. While such an SRP could potentially improve the system throughput, how to fully exploit this capability in practical systems is still questionable. In particular, the number of active users contending for the channel varies over time which complicates the algorithm design. By fully exploiting the potential of SIC, the proposed algorithm is designed to maximize the system throughput and minimize the access delay. For this purpose, an online estimation is introduced to estimate the number of active users in real-time and controls their transmissions accordingly. It is shown that the throughput of the proposed algorithm can reach up to 0.693 packets/slot under practical assumptions, which is the first result achieving the throughput limit proved by Yu-Giannakis. It is further shown that the system throughput of 0.559 packets/slot (80.6$\%$ of the throughput limit) is still achievable when the SIC capability is restricted by two.