Using the in silico experimental evolution platform Aevol, we have tested the existence of a "complexity ratchet" by evolving populations of digital organisms under environmental conditions in which simple organisms can very well thrive and reproduce. We observed that in most simulations, organisms become complex although such organisms are a lot less fit than simple ones and have no robustness or evolvability advantage. This excludes selection from the set of possible explanations for the evolution of complexity. However, complementary experiments showed that selection is nevertheless necessary for complexity to evolve, also excluding non-selective effects. Analyzing the long-term fate of complex organisms, we showed that complex organisms almost never switch back to simplicity despite the potential fitness benefit. On the contrary, they consistently accumulate complexity on the long term, meanwhile slowly increasing their fitness but never overtaking that of simple organisms. This suggests the existence of a complexity ratchet powered by negative epistasis: mutations leading to simple solutions, that are favourable at the beginning of the simulation, become deleterious after other mutations-leading to complex solutionshave been fixed. This also suggests that this complexity ratchet cannot be beaten by selection, but that it can be overthrown by robustness because of the constraints it imposes on the coding capacity of the genome.