This paper presents a novel nonlinear framework for the construction of flexible multivariate dependence structure~(i.e., copula) from existing copulas based on a straightforward ``pairwise max'' rule. The newly constructed max-copula has a closed form and has strong interpretability. Compared to the classical ``linear symmetric'' mixture copula, the max-copula can be viewed as a ``non-linear asymmetric'' framework. It is capable of modeling asymmetric dependence and joint tail behavior while also offering good performance in non-extremal behavior modeling. Max-copulas that are based on single-factor and block-factor models are developed to offer parsimonious modeling for structured dependence, especially in high-dimensional applications. Combined with semi-parametric time series models, the max-copula can be used to develop flexible and accurate models for multivariate time series. A new semi-parametric composite maximum likelihood method is proposed for parameter estimation, where the consistency and asymptotic normality of estimators are established. The flexibility of the max-copula and the accuracy of the proposed estimation procedure are illustrated through extensive numerical experiments. Real data applications in Value at Risk estimation and portfolio optimization for financial risk management demonstrate the max-copula's promising ability to accurately capture joint movements of high-dimensional multivariate stock returns under both normal and crisis regime of the financial market.