In this Wikipedia article some upper bounds for the smallest primitive root $g$ modulo a prime $p$ are given, but the first is implicite (what is the constant $C$ depending on $\epsilon$) and the other is only proven for ridiculous large primes.
What is the best known unconiditionally proven explicite upper bound that holds from a reasonably large prime on, say for $p\ge 10^6$ ?
Jana Pretorius posted a preprint that contains Theorem 2: the least primitive root modulo $p$ is less than $p^{0.68}$ for all primes $p$. (Note that one cannot do too much better with this exact shape of statement, since $2\approx 3^{0.631}$.) It might be possible to do better using Theorem 3 of this paper of McGown and Trudgian, although probably not since Pretorius acknowledges McGown and Trudgian in their preprint, suggesting that they wouldn't have produced redundant results.