making new symbol from two symbols

I would use \mkern instead of \kern, because it uses unit mu, which follows the math style:

\documentclass{article}
\newcommand*{\doublenabla}{%
  {\nabla\mkern-12mu\nabla}% subformula acts as \mathord
}
\begin{document}
\[ \doublenabla_{\doublenabla_{\doublenabla}} \]
\end{document}

Adjust the shift value to your needs.

Another possibility using mathtools to specify the offset from the left rather than from the right produces slightly different spacing:

\documentclass{article}
\usepackage{mathtools}
\newcommand{\nnabla}{\mathrlap{\nabla}\,\nabla}
\begin{document}
$\nnabla_{\nnabla_{\nnabla}}$
\end{document}