Balls into bins: explanation for $n=2$

Question

I discovered that the maximum number of balls in any bin in a random allocation of $n$ balls into $n$ bins is equal to $$\dfrac{\log n}{\log\log n},$$ with probability $1-1/n$. How to interpret this for fixed $n$? For example, for $n=2$, what does it mean to have $$\dfrac{\log 2}{\log\log 2}=-1.89\ldots$$