I had wondered about this too a long time ago and then came across this. The series
$$\sum_{n=3}^{\infty}\frac{1}{n\ln n (\ln\ln n)}=\infty$$
diverges and it can be very easily proven by the integral test. But here is the kicker. This series actually requires googolplex number of terms before the partial sum exceeds 10. Talk about slow! It is only natural that if the natural log is slow, then take the natural log of the natural log to make it diverge even slower.
Here is another one. This series
$$\sum_{n=3}^{\infty}\frac{1}{n\ln n (\ln\ln n)^2}=38.43...$$
actually converges using the same exact (integral) test. But it converges so slowly that this series requires $10^{3.14\times10^{86}}$ before obtaining two digit accuracy. So using these iterated logarithms you can come up with a series which converges or diverges "arbitrarily slowly".
Reference: Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, 1996.
