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 to 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.  

As pointed out below, you can compute $c_i$ mod $p$ up to $c_p.$ If any of these is $0 \bmod p$ you can stop there. Otherwise, if $c_p+d =0 \bmod p$ then problems if any, will be before $p+1$ or after, but not there.