Using the Aevol digital genetics platform we designed an in silico experiment to study the relationship between molecular complexity and phenotypic complexity: We evolved populations of digital organisms in an environment designed to enable survival of the simplest possible organism: one which genome encodes a single gene. By repeatedly evolving populations in this experimental design, we observed that ~1/3 of the lineages quickly found this simple genotype and were then stable for the rest of the experiment. But most lineages were not able to find this simple solution and showed slow gene acquisition along the 100,000 generations of the experiment. Importantly, simple organisms ended up with a very high fitness while complex genotypes ended up with a ~10x lower fitness. This shows that, even in a simple environment, evolution leads to a “complexity ratchet”: each gene acquisition creates the potential for the acquisition of further genes, ultimately pushing evolution towards complex solutions even in a simple environment. Moreover organisms engaged in this process were never able to outcompete the simple ones, showing that selection is not able to invert the complexity ratchet.