Digital assets termed cryptocurrencies are creating new paradigms for financial transactions as well as alternative means of capital. Despite their increasing relevance in financial situations, a complete understanding of the entire system remains lacking. Here the cryptocurrency market is treated as a complex system and analyzed using methods from statistical physics. The complexity-entropy causality plane (or CH plane) is employed in order to explore disorder and complexity in the space of cryptocurrencies. They are found to exist on distinct planar locations in the representation space, ranging from structured to stochastic-like behavior. The temporal trajectories of entropy and statistical complexity of prices vary drastically with position along the plane. Lastly cryptocurrencies appear to be characterized by ordinal patterns that represent strictly decreasing or increasing trends in price. The present analysis expands the understanding of and helps to quantify varying degrees of complexity in cryptocurrencies.