Is there a way to determine how many times a given cell divided?
Yes, its called the HayFlick limit. A cell will divide until their telomeres reach a critical length,which is usually around 40 to 60 cell divisions. The cells then enter cellular senescence and cease to divide. However, cells may also stop dividing due to (ROS) or reactive oxygen species, cell to cell fusions, or activation of oncogenes, the latter of which is a gene that has the potential to cause cancer.