Compression is essential to storing and transmitting medical videos, but the effect of compression artifacts on downstream medical tasks is often ignored. Furthermore, systems in practice rely on standard video codecs, which naively allocate bits evenly between medically interesting and uninteresting frames and parts of frames. In this work, we present an empirical study of some deficiencies of classical codecs on gastroenterology videos, and motivate our ongoing work to train a learned compression model for colonoscopy videos, which we call ``GastroEnterology Aware Compression" (GEAC). We show that H264 and HEVC, two of the most common classical codecs, perform worse on the most medically-relevant frames. We also show that polyp detector performance degrades rapidly as compression increases, and explain why a learned compressor would degrade more gracefully. Many of our proposed techniques generalize to medical video domains beyond gastroenterology.