Let's define the relatively symmetric group $RS(n, m)$, as the relatively free group in ${\rm Var}(S_n)$ on $m$ generators. Is it known, for what $n$ and $m$ is $RS(n, m)$ finite?
For $m = 1$ it always does, as $RS(n, m)$ is a cyclic group of finite exponent. For $n < 4$ it is also true, because any finitely generated group of exponents $1$, $2$ or $6$ is finite. This however, already does not work with $n = 4$, as the exponent of $S_4$ is $12$.
Any variety generated by a finite algebraic structure is locally finite. This means that the finitely generated relatively free algebras in any finitely generated variety are finite. Thus, $RS(m,n)$ is finite for any finite $m$ and $n$.