As the demand for high-quality data in model training grows, researchers and developers are increasingly generating synthetic data to tune and train LLMs. However, current data generation methods rely on seed sets containing tens of thousands of examples to prompt instruction-tuned models. This reliance can be especially problematic when the curation of high-quality examples is expensive or difficult. In this paper we explore the novel few-shot synthetic data generation setting -- generating a high-quality dataset from a few examples. We show that when working with only a few seed examples, instruction-tuned models used in current synthetic data methods produce insufficient diversity for downstream tasks. In contrast, we show that base models without post-training, largely untapped for synthetic data generation, offer substantially greater output diversity, albeit with lower instruction following abilities. Leveraging this insight, we propose Base-Refine (BARE), a novel two-stage method that combines the diversity of base models with the quality assurance of instruction-tuned models. BARE excels in few-shot synthetic data generation: using only 3 seed examples it generates diverse, high-quality datasets that significantly improve downstream task performance. We show that fine-tuning Llama 3.1 8B with 1,000 BARE-generated samples achieves performance comparable to state-of-the-art similarly sized models on LiveCodeBench tasks. Furthermore, data generated with BARE enables a 101% improvement for a fine-tuned Llama 3.2 1B on GSM8K over data generated by only instruction-models, and an 18.4% improvement for a fine-tuned Llama 3.1 8B over the state-of-the-art RAFT method for RAG data generation.