  Determining the approximate degree composition for Boolean functions remains
a significant unsolved problem in Boolean function complexity. In recent
decades, researchers have concentrated on proving that approximate degree
composes for special types of inner and outer functions. An important and
extensively studied class of functions are the recursive functions,
i.e.~functions obtained by composing a base function with itself a number of
times. Let $h^d$ denote the standard $d$-fold composition of the base function
$h$.
  The main result of this work is to show that the approximate degree composes
if either of the following conditions holds:
  \begin{itemize}
  \item The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of
the form $h^d$, with $h$ being any base function and $d= \Omega(\log\log n)$.
  \item The inner function is a recursive function of the form $h^d$, with $h$
being any constant arity base function (other than AND and OR) and $d=
\Omega(\log\log n)$, where $n$ is the arity of the outer function.
  \end{itemize}
  In terms of proof techniques, we first observe that the lower bound for
composition can be obtained by introducing majority in between the inner and
the outer functions. We then show that majority can be \emph{efficiently
eliminated} if the inner or outer function is a recursive function.
