“Is there a limit on how many times a research project (R01) grant can be renewed?”

No, there is not a rule that caps the number of times an R01 award can receive competitive renewal funding. As stated in our Advice for a Successful Renewal, competing renewal applications should link back to the previous grant's Specific Aims and show progress but not duplicate the Aims of the previous project period. So long as you choose an appropriate funding opportunity announcement and can demonstrate to reviewers the significance of your scientific questions and approach, you may apply to renew your research project’s funding support.