We study numerical approximations of stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities such as stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and spatially-extended stochastic FitzHugh-Nagumo equations. The exponential Euler scheme and the implicit-linear Euler scheme diverge strongly and numerically weakly in the case of some of such SPDEs. We introduce an appropriately stopped version of the exponential Euler scheme which we refer to as the nonlinearity-stopped exponential Euler scheme. This scheme stops the approximation process when the norm of the drift nonlinearity reaches some larger value than the reciprocal of the corresponding time step size. We show that the nonlinearity-stopped exponential Euler scheme converges strongly in the case of several SPDEs with superlinearly growing nonlinearities. Joint work with Martin Hutzenthaler and Arnulf Jentzen.