Are there good upper and lower estimates of $$ A(x):=\sum_{p \leq x} \log \log p $$ ignoring the first negative term?
Magma says
sum until 100 is 28.55
sum until 1000 is 286.41
sum until 10000 is 2544.87
sum until 100000 is 22382.24
sum until 1000000 is 199280.38
sum until 10000000 is 1799506.07
and the growth seems faster than a fixed power of log x.
Using $p_n\sim n\log n$,
$$A(n\log n)\sim\sum_{k=3}^n\log\log(k\log k)\sim\sum_{k=3}^n\log\log k.$$
Then from
$$\int\log\log x\,dx=x\log\log x-\text{Li}(x)$$
and
$$\text{Li(x)}\sim\frac x{\log x}$$ making this term negligible, we draw
$$\sum_{k=3}^n\log\log k\sim n\log\log n.$$
Now, with $n\log n=m$,
$$m\sim\frac n{\log n}$$ and finally
$$A(m)\sim\frac m{\log m}\log\log\frac m{\log m}\sim\frac{m\log\log m}{\log m}.$$
Due to the fact that the summand is growing extremely slowly, I guess that empirical observation of the asymptotic behavior is hopeless, though the agreement to the given data looks fair.