Continual learning in transformers is commonly addressed through parameter-efficient adaptation: prompts, adapters, or LoRA modules are specialized per task while the backbone remains frozen. Although effective in controlled multi-epoch settings, these approaches rely on gradual gradient-based specialization and struggle in Online Continual Learning (OCL), where data arrive as a non-stationary stream and each sample may be observed only once. We recast continual learning in transformers as a routing problem: under strict online constraints, the model must dynamically select the appropriate representational subspace for each input without explicit task identifiers or repeated optimization. We thus introduce Routing without Forgetting (RwF), a transformer architecture augmented with energy-based associative retrieval layers inspired by Modern Hopfield Networks. Instead of storing or merging task-specific prompts, RwF generates dynamic prompts through single-step associative retrieval over the transformer token embeddings at each layer. Retrieval corresponds to the closed-form minimization of a strictly convex free-energy functional, enabling input-conditioned routing within each forward pass, independently of iterative gradient refinement. Across challenging class-incremental benchmarks, RwF improves over existing prompt-based methods. On Split-ImageNet-R and Split-ImageNet-S, RwF outperforms prior prompt-based approaches by a large margin, even in few-shot learning regimes. These results indicate that embedding energy-based associative routing directly within the transformer backbone provides a principled and effective foundation for OCL.