Large Language Models (LLMs) have a natural role in answering complex queries about data streams, but the high computational cost of LLM inference makes them infeasible in many such tasks. We propose online cascade learning, the first approach to addressing this challenge. The objective here is to learn a "cascade" of models, starting with lower-capacity models (such as logistic regressors) and ending with a powerful LLM, along with a deferral policy that determines the model that is used on a given input. We formulate the task of learning cascades online as an imitation-learning problem and give a no-regret algorithm for the problem. Experimental results across four benchmarks show that our method parallels LLMs in accuracy while cutting down inference costs by as much as 90%, underscoring its efficacy and adaptability in stream processing. Anonymized source code is available at https://anonymous.4open.science/r/onlinecascadelearning.