Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.