  We examine the conjugacy growth series of all wreath products of the finitary
permutation groups $\text{Sym}(X)$ and $\text{Alt}(X)$ for an infinite set $X$.
We determine their asymptotics, and we characterize the limiting behavior
between the $\text{Alt}(X)$ and $\text{Sym}(X)$ wreath products. In particular,
their ratios form a limit if and only if the dimension of the symmetric wreath
product is twice the dimension of the alternating wreath product.
