Repetitive action counting, which aims to count periodic movements in a video, is valuable for video analysis applications such as fitness monitoring. However, existing methods largely rely on handcrafted models with limited representational capacity, which hampers their ability to accurately capture variable periodic patterns. Additionally, their supervised learning on narrow, limited training sets leads to overfitting and restricts their ability to generalize across diverse scenarios. To address these challenges, we propose CountLLM, the first large language model (LLM)-based framework that takes video data and periodic text prompts as inputs and outputs the desired counting value. CountLLM leverages the rich clues from explicit textual instructions and the powerful representational capabilities of pre-trained LLMs for repetitive action counting. To effectively guide CountLLM, we develop a periodicity-based structured template for instructions that describes the properties of periodicity and implements a standardized answer format to ensure consistency. Additionally, we propose a progressive multimodal training paradigm to enhance the periodicity-awareness of the LLM. Empirical evaluations on widely recognized benchmarks demonstrate CountLLM's superior performance and generalization, particularly in handling novel and out-of-domain actions that deviate significantly from the training data, offering a promising avenue for repetitive action counting.