A better question might be for which $m$ is there a sequence which fails at $c_{m+1}.$ That might be hard to answer because these sequences grow very quickly. Failure is much more likely when $m=p$ is a prime. Naively, the chance of failure in a case like this is about $\frac1{e}$ so there is likely to be a failure eventually. If you start with $c_2=kt!$ then you are sure do get past $t$ steps.
Maybe you can adjust $c_2$ to get to the prime you want. But it would be hard to investigate numerically.