In this paper, we consider a mixed-prompt scenario for a large language model (LLM) inference serving system that supports diverse applications with both short prompts and long prompts and heterogeneous SLOs for iteration time. To improve throughput when handling long prompts, previous research introduces a chunking method, but has not addressed heterogeneous SLOs. To address the limitation, we propose AccelGen, a high-throughput LLM inference serving system with heterogeneous SLO guarantees for diverse applications. AccelGen introduces four core components: (1) SLO-guaranteed dynamic chunking, which dynamically adjusts chunk sizes to maximize GPU compute utilization while meeting iteration-level SLOs; (2) Iteration-level SLO-based task prioritization, which prioritizes tight-SLO requests and batches requests with similar SLOs; (3) Multi-resource-aware batching, which selects queued requests to maximize the utilizations of both GPU compute resource and key-value cache (KVC). Trace-driven real experiments demonstrate that AccelGen achieves 1.42-11.21X higher throughput, 1.43-13.71X higher goodput, 37-90% higher SLO attainment, and 1.61-12.22X lower response latency compared to the state-of-the-art approaches. It achieves performance near the Oracle, which optimally maximizes goodput.