Classifying scanned documents is a challenging problem that involves image, layout, and text analysis for document understanding. Nevertheless, for certain benchmark datasets, notably RVL-CDIP, the state of the art is closing in to near-perfect performance when considering hundreds of thousands of training samples. With the advent of large language models (LLMs), which are excellent few-shot learners, the question arises to what extent the document classification problem can be addressed with only a few training samples, or even none at all. In this paper, we investigate this question in the context of zero-shot prompting and few-shot model fine-tuning, with the aim of reducing the need for human-annotated training samples as much as possible.