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Let $p_n$ be the $n$-th prime, then from Wikipedia I got that

$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.

What is a better approximation that includes $O\big(\frac{(\ln \ln n)^3}{\ln^3 n}\big)$?

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You can find an in-depth answer to your question in this paper of de Reyna and Jeremy. See in particular (65)-(66) along with (30) and Theorem 4.9. See also Theorem 6.2.

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