In quantitative magnetic resonance T1 mapping, the Variable Flip Angle (VFA) steady state spoiled gradient recalled echo (SPGR) imaging technique is popular as it provides a series of high resolution T1 weighted images in a clinically feasible time. Fast, linear methods that estimate T1 maps from these weighted images have been proposed, such as DESPOT1 and iterative reweighted linear least squares (IRWLLS). More accurate, nonlinear least squares (NLLS) estimators are in play, but these are generally much slower and require careful initialization. In this work, we present NOVIFAST, a novel NLLS-based algorithm specifically tailored to VFA SPGR T1 mapping. By exploiting the particular structure of the SPGR model, a computationally efficient, yet accurate and precise T1 map estimator is derived. Simulation and in vivo human brain experiments demonstrate a twenty-fold speed gain of NOVIFAST compared to conventional gradient-based NLLS estimators, while maintaining a high precision and accuracy. Moreover, NOVIFAST is eight times faster than efficient implementations of the VARPRO method. Furthermore, NOVIFAST is shown to be robust against initialization.