The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :

Is it known that there are infinitely many primes p for which the least prime q which is 1 mod p is > c p^2 (for some positive constant c, independent of p)?

Wikipedia's article on primes in arithmetic progressions says that the expected bound for the least prime is p^{2 + \epsilon}, given various strengthenings of the Riemann Hypothesis, but it doesn't say much about lower bounds.

By the way, for the applications in the linked post, it would be even better if the same lower bound applied to finding a prime power q which is 1 mod p.