We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consists of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. This permits statistical modelling of the uncertainty in a variety of algorithms that exploit first and second order Taylor expansions. To demonstrate the utility of this Gaussian process model we introduce new probabilistic versions of the classical extended Kalman filter for non-linear state estimation and the Euler method for solving ordinary differential equations.