How to typeset this matrix equation using simple LaTeX tools

Nest the second bmatrix as the first element of a matrix environment and add to the latter two empty rows:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation*}
  \begin{bmatrix}
     \\
   \; v\; \\
     \\
  \end{bmatrix}\!\!
\begin{matrix}
  \begin{bmatrix}
  \; w\; \\
  \end{bmatrix}
 \\{}\\ {}
\end{matrix}
\end{equation*}

\end{document} 

Let TeX do the computations.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation*}
A=
\vcenter{\hbox{\valign{\hbox{$#$}\vfil\cr
  \begin{bmatrix} \mbox{} \\ \quad v \quad  \\ \mbox{} \end{bmatrix} \cr
  \begin{bmatrix} \quad w \quad \end{bmatrix} \cr
}}}
\end{equation*}

\end{document}

Perhaps nesting array-environments and \vphantom in some places does the trick—but I assume there are much better solutions.

\documentclass{article}
\usepackage{array}

\begin{document}
$A=\left[%
     \begin{array}{c}%
       {\vphantom{\left[\begin{array}{c}w\end{array}\right]}}\\
       {v}\\
       {\vphantom{\left[\begin{array}{c}w\end{array}\right]}}\\
     \end{array}%
   \right]\mathclose{}\  %
   \kern-\arraycolsep
   \begingroup
   % Adjust \arraycolsep to regulate the horizontal gaps between "[" and "w" and "]"
   \arraycolsep=1\arraycolsep\relax
   \kern-\arraycolsep
   \begin{array}{c}%
     {\mathopen{}\left[\begin{array}{c}w\end{array}\right]}\\
     {\vphantom{v}}\\
     {\vphantom{\left[\begin{array}{c}w\end{array}\right]}}\\
   \end{array}\kern-\arraycolsep\endgroup=A$
\end{document}