Human brain cortex acts as a rich inspiration source for constructing efficient artificial cognitive systems. In this paper, we investigate to incorporate multiple brain-inspired computing paradigms for compact, fast and high-accuracy neuromorphic hardware implementation. We propose the TripleBrain hardware core that tightly combines three common brain-inspired factors: the spike-based processing and plasticity, the self-organizing map (SOM) mechanism and the reinforcement learning scheme, to improve object recognition accuracy and processing throughput, while keeping low resource costs. The proposed hardware core is fully event-driven to mitigate unnecessary operations, and enables various on-chip learning rules (including the proposed SOM-STDP & R-STDP rule and the R-SOM-STDP rule regarded as the two variants of our TripleBrain learning rule) with different accuracy-latency tradeoffs to satisfy user requirements. An FPGA prototype of the neuromorphic core was implemented and elaborately tested. It realized high-speed learning (1349 frame/s) and inference (2698 frame/s), and obtained comparably high recognition accuracies of 95.10%, 80.89%, 100%, 94.94%, 82.32%, 100% and 97.93% on the MNIST, ETH-80, ORL-10, Yale-10, N-MNIST, Poker-DVS and Posture-DVS datasets, respectively, while only consuming 4146 (7.59%) slices, 32 (3.56%) DSPs and 131 (24.04%) Block RAMs on a Xilinx Zynq-7045 FPGA chip. Our neuromorphic core is very attractive for real-time resource-limited edge intelligent systems.