Agentic applications are shifting AI serving from isolated model inference to long-running workloads in which LLMs coordinate tools, environments, and persistent state. However, the system behavior of these workloads---where latency, cost, and bottlenecks arise---remains poorly characterized, leaving serving systems to rely on assumptions built for conventional inference. We present AgentSysBench, a benchmark suite and measurement toolkit with ten representative agentic applications and unified systems-level instrumentation. Across controlled deployments and production traces, we identify six properties that distinguish agentic workloads from conventional LLM serving: (1) execution is heavyweight and stateful, with non-LLM components dominating latency in 5 of 10 applications and sandbox working-set memory peaking at 28 GB per session; (2) applications compose components with heterogeneous resource affinity---GPU-bound inference, memory-bound retrieval, CPU-bound sandboxes---whose task latencies diverge by up to 32x; (3) bottlenecks shift across requests, models, and deployments; (4) production sessions hold state idle for minutes to hours between active steps; (5) a control-plane tax---auxiliary LLM calls and context overhead from tool schemas and observations---crowds out productive compute and context; and (6) production traces from three applications reveal heavy cross-request redundancy in search queries and web fetches, exposing a large caching opportunity. Four design explorations demonstrate that these findings are actionable: task-aware serving reduces latency by 29--40%, communication-aware placement by up to 4.5x, state offloading reduces memory usage by 4.6x, and tool-result caching removes 35.2% of redundant search calls and saves 19.3% of aggregate search latency.