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there is no hard limit, but its a question of probability.
cance: epithelial cells are exposed to the most mutagens, so they get the most cancer, but all cells can get cancer. it is basicly evolution. all bodily cells have a mutation rate, and some mutations (preoncogenic genes etc) cause them to live longer, which is an evolutionary edge.
aging: there are about 5000 genes that guide the process, but they dont cause it. disrupting them has resaulted in modell organism living 6x normal lifespan (c elegans)
aging is mostly caused by transposons, these also cause cancer and neurodegeneration. these are bits of dna (old, integrated, degraded viral dna) bordered by inverted terminal repeats, and some of them still code for proteins that cut or copy and past them all over your dna. about 67% of your dna consists of transposons. these are kept dormant by piwi proteins. every mitosis caryes a chance of one of them getting lose. as more and more of them get lose, more and more mutations occure.
telomeres: the little cap on the end of your dna, also protects it from degradation and mutation. the more often your celles devide, the more these degrade, the higher the cells mutation rate.
to summarise, cells have mutation rates, decreasing telomere length increases mutation rate, and there is a rate at which transposons become active, and cause mutations. mutations cause aging and death.