Is there an example of an i.i.d. sequence of random variables $X_n$ so that $S_n/n\to 0$ in probability but $S_n/n$ does not converge a.s.?
This post has an answer in which the sequences are independent but not i.i.d.
We know from the strong law of large numbers that if $EX_1=c\in [-\infty,\infty]$, then $S_n/n\to EX_1$ a.s.
Hence such an example must have $EX_1^+=\infty$ and $EX_1^-=\infty$.