How can I print three greater-than symbols after each other (>>>)?

The addition of a group around the central > will prevent each from seeing the others, and therefore should prevent (for lack of a better word) the ligature from forming.

Note: See comments to this answer, which are very illuminating, which explains why this technique will work in pdflatex but not lualatex!

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\begin{document}
\texttt{>{>}> versus >>>}
\end{document}

This happens when you use the T1 encoding.

You can disable the ligature that maps >> to » with microtype. I also removed the ligature from << to « for symmetry. The first line shows that the ligatures are not suppressed altogether, but only when using the monospaced font.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{courier}
\usepackage{microtype}

\DisableLigatures[<,>]{encoding=T1,family=tt*}

\begin{document}

<<This is quoted>>

\texttt{>>>}

\texttt{<<<}

\end{document}