Sum of logs $\log x + \log\log x +...$

Question

Is there a reduction for this infinite sum?

$$\log x + \log\log x + \log\log\log x +... = ?$$

for all $x > 0$?

Answer 1

The sum diverges with any reasonable interpretation of the logarithms. After all, $\log \log \cdots \log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.