  Let $f(x) \in \mathbb{Z}[x]$. Set $f_{0}(x) = x$ and, for $n \geq 1$, define
$f_{n}(x)$ $=$ $f(f_{n-1}(x))$. We describe several infinite families of
polynomials for which the infinite product \prod_{n=0}^{\infty} (1 +
\frac{1}{f_{n}(x)}) has a \emph{specializable} continued fraction expansion of
the form S_{\infty} = [1;a_{1}(x), a_{2}(x), a_{3}(x), ... ], where $a_{i}(x)
\in \mathbb{Z}[x]$, for $i \geq 1$. When the infinite product and the continued
fraction are \emph{specialized} by letting $x$ take integral values, we get
infinite classes of real numbers whose regular continued fraction expansion is
predictable. We also show that, under some simple conditions, all the real
numbers produced by this specialization are transcendental.
