The substantial volatility and growth in cryptocurrencies valuations between 2009 and the end of 2017 strongly suggest that both long memory and price volatility and return spillovers should be present in these assets’ dynamics. To date, literature on the major cryptocurrencies price processes does not address jointly and comprehensively their fractal properties, long memory and wavelet analysis, that could robustly confirm the presence of fractal dynamics in their prices, and confirm or deny the validity of the Fractal Market Hypothesis as being applicable to the cryptocurrencies. This research shows that Bitcoin prices exhibit long term memory, although its trend has been reducing overtime. In fact, assessing Bitcoin, Ethereum and Ripple across the period between 2016 and 2017, focusing solely on the period prior to the crash of 2018, we can conclude that Bitcoin was better described by a random walk, showing signs of markets maturity emerging, in contrast, other cryptocurrencies such as Ethereum and Ripple present evidence of a growing underlying memory behaviour.