Two main lines of research link information theory to evolutionary biology. The first focuses on organismal phenotypes, and on the information that organisms acquire about their environment. The second connects information-theoretic concepts to genotypic change. The genotypic and phenotypic level can be linked by experimental high-throughput genotyping and computational models of genotype-phenotype relationships. I here use a simple information-theoretic framework to compute a phenotype's information content (its phenotypic complexity), and the information gain or change that comes with a new phenotype. I apply this framework to experimental data on DNA-binding phenotypes of multiple transcription factors. Low phenotypic complexity is associated with a biological system's ability to discover novel phenotypes in evolution. I show that DNA duplications lower phenotypic complexity, which illustrates how information theory can help explain why gene duplications accelerate evolutionary adaptation. I also demonstrate that with the right experimental design, sequencing data can be used to infer the information gain associated with novel evolutionary adaptations, for example in laboratory evolution experiments. Information theory can help quantify the evolutionary progress embodied in the discovery of novel adaptive phenotypes.