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We present variants of the conjugate gradient (CG), conjugate residual (CR), and generalized minimal residual (GMRES) methods which are both pipelined and flexible. These allow computation of inner products and norms to be overlapped with operator and nonlinear or nondeterministic preconditioner application. The methods are hence aimed at hiding network latencies and synchronizations which can become computational bottlenecks in Krylov methods on extreme-scale systems or in the strong-scaling limit. The new variants are not arithmetically equivalent to their base flexible Krylov methods, but are chosen to be similarly performant in a realistic use case, the application of strong nonlinear preconditioners to large problems which require many Krylov iterations. We provide scalable implementations of our methods as contributions to the PETSc package and demonstrate their effectiveness with practical …