Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the law of the iterated logarithm that there is no convergence almost sure, but-according to wikipedia- $$\frac{S_n}{\sqrt{n \log(\log(n))}} \rightarrow 0$$ in probability.
I don't know where this comes from. The laws of large numbers are to weak to show this. Therefore, I would appreciate it if anybody could explain, why this is true?
To expand on Did's comment:
By the central limit theorem, $\frac{S_n}{\sqrt{n}} \to N(0,1)$ in distribution. Now $\frac{1}{\sqrt{\log\log n}} \to 0$ (either as real numbers, or as constant random variables converging in probability), so by Slutsky's theorem $\frac{S_n}{\sqrt{n \log\log n}} \to 0$ in distribution. But a sequence converging in distribution to a constant also converges in probability.