I think such an exploration can be a great independent exploration for interested and capable students, and so I will expand a little on the question.
It is obvious, I think that extending 11 by adding more ones will quickly produce a multiple of three. And 1111 and 11111 as well as 1111111 are all composite. But extending 13 to 133, 1333, and 1333 are not prime. 17 undergoes a similar failure to produce primes. But 19, 199, and 1999 are all prime. Unfortunatly, 19999 is 7 x 2857.
With 23, 233, 2333, 23333 are again all prime. Is it possibe for this method to produce more than four primes in a row.
For more than two digits there are extended options. We could repeat the last digit, extending xyz to xyzz, or alternately replicating the last two digits, xyzyz.
For students working on divisibiity schemes, testing primes, etc this is probably great practice, and for others, it cold just be good fun.