LUNAR is a new benchmark for evaluating personalized large language models (LLMs) on universal user behavior logs. Existing personalized LLM benchmarks mainly rely on textual personas or isolated behavioral signals, limiting the evaluation of cross-domain behavioral personalization. In LUNAR, we use a multi-stage coarse-to-fine synthesis pipeline grounded in real-world behavioral patterns to support scalable benchmark construction. Fidelity analyses show that LUNAR's behavioral data is more closely aligned with real behavioral distributions. Experiments on 19 mainstream LLMs show that access to behavioral logs is necessary but not sufficient for deep personalization: neither more context nor larger models guarantees better performance; effective personalization depends on selecting and integrating relevant evidence across domains. Direct retrieval of fine-grained behavioral records consistently outperforms compressed memory, while stronger personalization can come at the cost of privacy protection. These findings identify evidence selection, cross-domain integration, and privacy control as key challenges for personalized LLMs. Blogger's Review: LUNAR benchmark provides a new perspective on evaluating personalized LLMs on universal user behavior logs, highlighting the importance of personalization and the challenges of privacy protection.