Test-time adaptation enables large language models (LLMs) to modify their behavior at inference without updating model parameters. A common approach is many-shot prompting, where large numbers of in-context learning (ICL) examples are injected as an input-space test-time update. Although performance can improve as more demonstrations are added, the reliability and limits of this update mechanism remain poorly understood, particularly for open-source models. We present an empirical study of many-shot prompting across tasks and model backbones, analyzing how performance varies with update magnitude, example ordering, and selection policy. We further study Dynamic and Reinforced ICL as alternative test-time update strategies that control which information is injected and how it constrains model behavior. We find that many-shot prompting is effective for structured tasks where demonstrations provide high information gain, but is highly sensitive to selection strategy and often shows limited benefits for open-ended generation tasks. Overall, we characterize the practical limits of prompt-based test-time adaptation and outline when input-space updates are beneficial versus harmful.