Numerically, one finds that the function
$$
  f(n) \ := \ \sum_{p \leq n, \\ p \ \text{prime}} \frac{1}{p}
  \left(1 - \frac{1}{\ln(\ln(p))}\right)
$$
appears to take its maximum at $n = 2$.
While already from taking a first glance it is clear that this cannot be, and that $f$ is
unbounded, I claim that it is numerically challenging to find the smallest $n$ for which we have
$f(n) > f(2)$ ... .