We introduce LORAKI, a novel MRI reconstruction method that bridges two powerful existing approaches (LORAKS and RAKI). Like RAKI (a deep learning extension of GRAPPA), LORAKI trains a scan-specific autocalibrated convolutional neural network (which only relies on autocalibration data, and does not require external training data) to interpolate missing k-space samples. However, unlike RAKI, LORAKI is based on a recurrent convolutional neural network architecture that is motivated by the iterated convolutional structure of a certain LORAKS algorithm. LORAKI is very flexible and can accommodate arbitrary k-space sampling patterns. Experimental results suggest LORAKI can have better reconstruction performance than state-of-the-art methods.