The permutation $\sigma$ with $\sigma(n) = n+1$ if $n$ is odd and $\sigma(n) = n-1$ if $n$ is even -- which you could represent as the infinite sequence of integers $(2,1,4,3,6,5,\ldots)$ -- is not a limit of finite permutations.
Added after the first two comments: it's been asked what I meant. I was thinking something like this: there does not exist a sequence $\sigma_1, \sigma_2, \sigma_3, \ldots$ with $\sigma_k \in S_k$, such that $\sigma_k$ is the restriction of $\sigma$ to $\{1, 2, \ldots, k \}$ for all but finitely many $k$. This notion of "limit" is not useful in any obvious way, though.