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He can continue to accelerate. Of course, as he can always claim to be "at rest", so what should stop him? But his velocity relative to someone else will never be greater than c.If the mass is the relativistic mass in dependent of the observer then for the moving observer he can continue to accelerate past c.
I don't think it makes sense for you to delve into GR before getting the basics of SR.Although the one regarding my flawed interpretation of g equation in GR needs addressing.