  We prove that the class of permutations generated by passing an ordered
sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series
is in bijection with an unambiguous context-free language, where a permutation
of length $n$ is encoded by a string of length $3n$. It follows that the
sequence counting the number of permutations of each length has an algebraic
generating function. We use the explicit context-free language to compute the
generating function: \begin{align*} \sum_{n\geq 0} c_n t^n &=
\frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q}
\end{align*} where $c_n$ is the number of permutations of length $n$ that can
be generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simple
variant of the Catalan generating function. This in turn implies that
$c_n^{1/n} \to 2+2\sqrt{5}$.
