How do I nest cases?

I'd not use the inner brace, but a standard notation for the minimum, just split across lines.

\documentclass{article}
\usepackage{amsmath}

\DeclareMathOperator{\lev}{lev}

\begin{document}

\begin{equation*}
\lev_{a,b}(i,j)=
  \begin{cases}
  \max(i,j) & \text{if $\min(i,j)=0$,} \\[1ex]
  \begin{aligned}[b]
  \min\bigl(\lev_{a,b}&(i-1,j)+1, \\
            \lev_{a,b}&(i,j-1)+1, \\
            \lev_{a,b}&(i-1,j-1)+1_{(a_i\ne b_j)}
      \bigr)
  \end{aligned} & \text{otherwise.}
\end{cases}
\end{equation*}

\end{document}

You might prefer the following realization, which is obtained by using \begin{aligned} instead of \begin{aligned}[b].

amsmath's cases is defined for this. The horizontal space between the bracket and the inside contents is very good.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
\mathrm{lev}_{a,b}(i,j)=\begin{cases}
    \max(i,j)&\text{if $\min(i,j)=0$,}\\
    \min\begin{cases}
        \mathrm{lev}_{a,b}(i-1,j)+1\\
        \mathrm{lev}_{a,b}(i,j-1)+1\\
        \mathrm{lev}_{a,b}(i-1,j-1)+1_{(a_i\ne b_j)}
    \end{cases} &\text{otherwise.}
\end{cases}
\]
\end{document}

---

It seems to me that you are writing a document which uses lev() function quite often. In this case, you should define a new macro to avoid repetition (I use \DeclareMathOperator, which is the best way for this, thanks to egreg's suggestion in his comment).

\documentclass{article}
\usepackage{amsmath}
%\newcommand{\lev}{\mathrm{lev}}: not good
\DeclareMathOperator{\lev}{lev}
\begin{document}
\[
\lev_{a,b}(i,j)=\begin{cases}
    \max(i,j)&\text{if $\min(i,j)=0$,}\\
    \min\begin{cases}
        \lev_{a,b}(i-1,j)+1\\
        \lev_{a,b}(i,j-1)+1\\
        \lev_{a,b}(i-1,j-1)+1_{(a_i\ne b_j)}
    \end{cases} &\text{otherwise.}
\end{cases}
\]
\end{document}

If you use the function lev a lot then it is a good idea to define it as follows

\newcommand{\lev}[2]{\mathrm{lev}_{a, \thinspace b} (#1, \thinspace #2)}

Just type \lev and you will get the function with two arguments to enter.

\documentclass{article}
\usepackage{amsmath}
\usepackage{booktabs}
\usepackage{array}

\begin{document}

\newcommand{\lev}[2]{\mathrm{lev}_{a, \thinspace b} (#1, \thinspace #2)}    
\begin{equation}
\lev{i}{j} = \left\lbrace
    \begin{array}{l l}
        \max(i, \thinspace j) & \text{if~} \min(i, \thinspace j) =0,
    \\
        \min \left\lbrace \hspace{-1mm}
                \begin{array}{l}
                    \lev{i-1}{j} + 1
                \\
                \addlinespace[0.5mm]
                    \lev{i}{j-1} + 1
                \\
                \addlinespace[0.5mm]
                    \lev{i-1}{j-1} + 1_{(a_{i} \neq b_{j})}
                \end{array}

            \right. & \text{otherwise}.
    \end{array}
            \right.
\end{equation}

\end{document}