In this paper, we study the contemporaneous tail dependence structure in a pairwise comparison of the ten largest cryptocurrencies, namely Bitcoin, Dash, Dogecoin, Ethereum, Litecoin, Monero, Namecoin, Novacoin, Peercoin, and Ripple. We apply multivariate extreme value theory and we estimate a bias-corrected extreme correlation coefficient. Our findings reveal clear patterns of significantly high bivariate dependency in the distribution tails of some of the most basic and widespread cryptocurrencies, primarily over various downside constraints. This means that extreme correlation is not related to cryptocurrency market volatility per se, but to the trend of the cryptocurrency market. Therefore, extreme correlation increases in bear markets, but not in bull markets for these pairs. Interestingly, there is also a significant number of pairs which exhibit a weak level of dependency in distribution tails.