Numeric lettrine is too small

the 0.3 here makes it 30% bigger which might be a bit too much, adjust to taste.

\documentclass{article}
\usepackage{lettrine}
\usepackage{libertine} % this is the font I'm using
\usepackage{lipsum}    % for generated text

\begin{document}
    \lettrine[loversize=0.3]{1}{}
    \lipsum[5] % generate sample text
\end{document}

The problem doesn't come from the font, but from the fact that \lettrine expects a second argument, which is by default in small caps. You can change that with \LettrineSecondString:

\documentclass{article}
\usepackage{lettrine}
\renewcommand*{\LettrineSecondString}{F}
\usepackage{libertine} % this is the font I'm using
\usepackage{lipsum} % for generated text
\begin{document}

    \lettrine[nindent=0.25em]{1}{}
    \lipsum[5] % generate sample text

\end{document} 

g

You can use a modified version of Bernard's answer (I cannot reproduce his output).

\documentclass{article}
\usepackage{lettrine}
\usepackage{libertine} % this is the font I'm using
\usepackage{lipsum} % for generated text

\renewcommand*{\LettrineSecondString}{\strut}
\setlength{\DefaultFindent}{0.5em}
\setlength{\DefaultNindent}{0pt}% this would be added to the findent

\begin{document}

\lettrine{1}{}\lipsum[5] % generate sample text

\end{document}

Note that you should leave no space after \lettrine{1}{}.

This way you need not guess appropriate values for loversize (and so not break parallel with XeLaTeX.