Intuitively, $\log n$ (base 2) is the number of times you have to divide $n$ by 2 before reaching a number around 2. (Waving our hands a little to gloss over floor vs ceiling and $\pm$ 1 errors.) Similarly, $\log \log n$ is the number of times we have to iterate the square root function to get from $n$ down to a number around $2$ (waving our hands again).
What is the intuition for $\log \log \log n$? What is the function that you would iterate $\log \log \log n$ times to get from $n$ down to a small constant?