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.
Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers
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 $EX_1^-=\infty$.