In this article we show the convergence of the operator norm (the largest singular value) of any polynomial in some random matrices. If X_N= (X_1^(N), ...,X_p^(N)) denotes a family of N-by-N independent random matrices from the Gaussian Unitary Ensemble (GUE), then we state sufficient conditions on (possibly random) matrices Y_N =(Y_1^(N), ..., Y_q^(N)), independent with X_N, for which ||P(X_N, Y_N, Y_N^*)|| converges for all polynomial P. Limits are described by operator norms of objects from free probability theory. We give examples of diagonal matrices $\mathbf Y_N $ for which the convergence holds. The case of block matrices is also investigated. Convergence of the operator norm holds when the GUE matrices are replaced by some non-white Wishart matrices.