I just created the following "game":
Choose some prime number. If it has $k$ digits then add another $k$ digits to the right to obtain another prime number. Now we have a prime number with $2k$ digits. Now add $2k$ digits to the right of the new number to obtain a new prime number. Repeat the procedure until it is no longer possible to produce new primes in this way.
To clarify, let us take for example $5$. We must add one digit to the right to obtain another prime number, it can be $53$. Now we must add two digits to the right of $53$ to obtain a new prime. It could be $5323$. Now we must add four digits to the right of $5323$ to obtain a new prime. It could be $53231117$ and so on and so on...
Is it true that whatever prime we choose at the beginning that this game never ends? That is, that it is always possible to build larger primes from the smaller ones in this way?