The proliferation of civilian and commercial unmanned aerial vehicles (UAVs) has heightened the demand for reliable radio frequency (RF)-based drone identification systems that can operate under dynamic and uncertain airspace conditions. Most existing RF-based recognition methods adopt a closed-set assumption, where all UAV types are known during training. Such an assumption becomes unrealistic in practical deployments, as new or unknown UAVs frequently emerge, leading to overconfident misclassifications and inefficient retraining cycles. To address these challenges, this paper proposes a unified incremental open-set learning framework for RF-based UAV recognition that enables both novel class discovery and incremental adaptation. The framework first performs open-set recognition to separate unknown signals from known classes in the semantic feature space, followed by an unsupervised clustering module that discovers new UAV categories by selecting between K-Means and Gaussian Mixture Models (GMM) based on composite validity scores. Subsequently, a lightweight incremental learning module integrates the newly discovered classes through a memory-bounded replay mechanism that mitigates catastrophic forgetting. Experiments on a real-world UAV RF dataset comprising 24 classes (18 known and 6 unknown) show effective open-set detection, promising clustering performance under the evaluated noise settings, and stable incremental adaptation with minimal storage cost, supporting the potential of the proposed framework for open-world UAV recognition.