This seems to be obvious and already known but at this moment I do not see an obvious answer.
Choose some prime number that has $d_1$ digits, so we start with a prime $a_1...a_{d_1}$.
Now the task is to build a larger prime from this prime by concatenating some digits to $a_1...a_{d_1}$ either from left or from right or both from left and from right. Then we get new, larger prime. The goal is to proceed further as much as possible.
Does this procedure never ends no matter with which prime we start?
This is the weakest of all the problems that I know of about concatenating digits to obtain primes so I will not be surprised if it has an easy and obvious answer.
Observe that there is no limit imposed on the number of digits that we can concatenate either from left or from right or both from left and from right.
Starting with any prime besides $2$ or $5$, we can always continue the process indefinitely by adding digits only to the left. Indeed, if $p$ is a $d$-digit prime number different from $2$ or $5$, then by Dirichlet's theorem there are infinitely many primes of the form $10^dn+p$, and any such prime is obtained from $p$ by adding digits to the left.
If we start with $2$ or $5$, we can go to $23$ or $53$ and thereafter only add digits to the left.