What is the correct way to typeset higher order time derivatives in LaTeX?

There's probably no single or unique "correct" way to typeset this expression, but

• using \scriptscriptstyle for the overset terms (4) and (n+1),

• using \mathclap to "smash" the width of the term (n+1), and

• inserting a thinspace, \,, in front of the dt terms

may be what you need:

\documentclass{article}
\usepackage{mathtools} % for \mathclap macro
\begin{document}
\[
 \dot{V}_t = \dot{V}_{t-1} + 
 \ddot{V}_{t-1} \,dt + 
 \frac{\dddot{V}_{t-1}\,dt^2}{2} +
 \frac{\overset{\scriptscriptstyle(4)}{V}_{t-1}\,dt^3}{6} + \dots + 
 \frac{\overset{\mathclap{\scriptscriptstyle(n+1)}}{V}_{t-1}\,dt^n}{n!}
\]
\end{document} 

I would just put them as ordinary superscripts. For the third derivative it seems to be necessary to backspace the subscript a bit:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{equation*}
  \dot{V}_t = \dot V_{t-1} + \ddot V_{t-1} dt +
  \frac{\dddot V_{\!t-1}dt^2}{2} + \frac{V^{(4)}_{t-1}dt^3}{6} +
  \dots + \frac{V^{(n+1)}_{t-1}dt^n}{n!}
\end{equation*}

\end{document}

If, as in your case, this notation is used for something else than I would recommend using a differential operator instead:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{equation*}
  \dot{V}_t = \dot V_{t-1} + \ddot V_{t-1} dt +
  \frac{(D^3_t V)_{t-1}dt^2}{2} + \frac{(D^4_tV)_{t-1}dt^3}{6} +
  \dots + \frac{(D^{n+1}_tV)_{t-1}dt^n}{n!}
\end{equation*}

\end{document}

Just to complete the answers of Mico and Andrew:

In Wikipedia, the dot notation for derivatives higher than the third (i.e., \ddot{V}   ) is \overset{n}{\dot{V}}

 

(that is, the n-th dot derivative)

Example:

(code: \overset{5}{\dot{V}} \equiv \frac{d^5V}{dt^5})