Extending primes This question is more of a curiosity than anything.
Start with a prime number and consider concatenating digits onto the right hand side. Sometimes you can make a prime and continue the process making bigger primes but this process always ends.
I have checked this computationally and find the biggest such prime created is the number $73939133$.
My question is, does there exist a theoretical explanation as to why this process ends? I can't find one myself but I am convinced there should be a reason.
 A: Here is a very rough argument.
The average gap between primes of size about $p$ is about $\ln p$. 
To add another digit to $p$ you have to find a prime amongst $10p+1$, $10p+3$, $10p+7$, $10p+9$. If there is a gap at least between $10p-1$ and $10p+11$ you won't be able to proceed. 
Obviously your process is cherry-picking the best examples, but even so, as the gaps increase in size you are more likely to find yourself in the middle of a gap as on the edge (with a prime to pick).
It would take a special feature of the distribution of primes to suggest that the process of adding digits in base $10$ (which has no obvious reason to be special) could go on for ever.
A: Between $N$ and $N+10$ there are approximately $\frac{10}{\ln N}$ primes. So if there are $a_n$ primes with $n$ digits (either starting at this length or counting only those obtained step by step from one-digit primes), we count primes in intervals $[N,N+1]$ for $a_n$ distinct values of $N$ and each is $\approx 10^n$. Then in the next round we can expect $a_{n+1}\approx a_n\cdot \frac{10}{n\ln 10}$ primes by appending a digit. As $n$ grows, this quickly brings $a_n$ down to zero.
