<p>Logo classification has garnered considerable attention due to its various applications, such as copyright protection, product recommendations, and contextual advertising. However, the rapid emergence of new companies and logo designs poses challenges for existing logo classification models, particularly under only limited examples for novel classes. To address this, the task of logo few-shot class-incremental learning (Logo-FSCIL) has been proposed, aiming to incrementally learn and recognize new logo knowledge while retaining the ability to discriminate among previously learned ones. Inspired by cognitive science, which highlights compositional learning as a core human ability, we leverage the inherent compositional nature of logos where visual primitives can be reused and recombined to represent new concepts, to enhance incremental learning in a more interpretable and effective manner. In this work, we propose ComAdPro, a compositional learning strategy with prototype adaptation tailored for the Logo-FSCIL task. Specifically, we design an optimized prototype-supervised compositional learning network that leverages both global and local features while mitigating catastrophic forgetting. Our approach includes a two-stage prototype optimization module to strengthen global feature representations, a multi-scale adapter to improve the alignment between visual components and prototypes, and a class-wise weighting scheme to refine joint predictions. Extensive experiments conducted on three logo-specific datasets and three general benchmarks demonstrate the effectiveness and generalizability of our approach.</p>