We provide a new class of observers for a class of nonlinear systems which are not required to be affine in the unmeasured states. The observers ensure exponential convergence of the observation errors to zero, under linear output measurements. The rate of exponential convergence converges to infinity, as the growth rate of the nonlinear statedependent part of the dynamics converges to zero, so we call the observers almost finite-time. Under global Lipschitz conditions on the state-dependent part of the dynamics, our global result ensures convergence of the observers, for all initial states. For cases where the nonlinearity is of order two at the origin, we provide local results ensuring exponential convergence of the observation errors to zero, when the initial state is small enough. We apply the results to a model of a pendulum with friction, and to dynamics with Lotka-Volterra nonlinearities.