Spiking neural networks (SNNs) promise energy-efficient computing due to their sparse spatio-temporal activity. However, effectively translating such irregular sparsity into practical performance and energy gains remains challenging, as full-event computing architectures are still underexplored. This paper proposes ExSpike, a general full-event neuromorphic architecture that fully exploits irregular sparsity in SNNs. To realize pure event-driven execution, we first propose a set of dataflow optimizations to ensure that the inputs to each SNN layer remain spike-based, thereby enabling full-event execution throughout the network. We then design a hardware-efficient full-event architecture, named ExSpike, which supports the optimized pure event-driven dataflow and an additional Attention Core for spike-driven self-attention. To further improve computing efficiency, we introduce adjacent-position event compression to reduce redundant accumulations across spatially adjacent spike sequences. ExSpike is implemented on an AMD Xilinx Virtex-7 FPGA and evaluated on both classification and segmentation workloads. Experimental results show that ExSpike achieves high normalized energy efficiency across diverse SNN models while maintaining competitive accuracy, delivering up to 479.15 GOPS, 281.85 GOPS/W, and 0.80 GOPS/W/PE. In particular, ExSpike achieves up to 10$\times$ higher PE-normalized energy efficiency than the SOTA FPGA-based SNN accelerator (FireFly-T). The code for ExSpike is available at https://github.com/xiaoyuehai/ExSpike.

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