A system for additively manufacturing a composite part is disclosed. The system may include a support moveable in multiple dimensions, and a print head connected to an end of the support. The print head may have a first matrix reservoir configured to wet a continuous strand reinforcement, a first nozzle fluidly connected to the first matrix reservoir, and a first cure enhancer located to expose the continuous strand reinforcement to cure energy upon discharge from the nozzle. The system may also include an in-situ fiber-making apparatus configured to supply the continuous strand reinforcement to the matrix reservoir of the print head.