Lettrine with EB Garamond Initials

The problem is that EB Garamond initials have a depth, which is about 40% of the height. At 10pt font size, the height of L is 7.39227pt, its depth is 2.87pt.

This shows why, when you reserve five lines for the height, two lines are overlapped.

You can play with loversize and lraise:

\documentclass{article}

\usepackage[latin]{babel}% just to avoid overfull boxes

\usepackage{fontspec}

\usepackage{lettrine}

\usepackage{lipsum}

% fonts
\setmainfont{EB Garamond}
\newfontface{\initials}{EB Garamond Initials}

% lettrine settings
\setcounter{DefaultLines}{5}
\renewcommand{\DefaultLraise}{0.25}
\renewcommand{\DefaultLoversize}{-0.25}
\renewcommand{\LettrineFontHook}{\initials}


\begin{document}

\lettrine{L}{orem ipsum} dolor sit amet, consectetuer adipiscing elit. \lipsum[2]

\end{document}

The following code after \begin{document} will print the dimensions for the letter L in EB Garamond Initials:

{\initials\xdef\initialsfont{\the\font}}
\noindent
Height: \the\fontcharht\initialsfont`L \\
Depth: \the\fontchardp\initialsfont`L

It is possible to add lines with the parameter depth=2 . Here, the space below becomes too large, so you can decrease the size of the lettrin with the lowersize=-0.1 parameter and depth=1, which leaves an extra space above that you decrease with lraise=0.1.

\documentclass{article}

\usepackage{fontspec}
\newfontface{\initials}{EB Garamond Initials}
\setmainfont{EB Garamond}

\usepackage{lettrine}
\setcounter{DefaultLines}{5}
\renewcommand{\LettrineFontHook}{\initials}

\usepackage{lipsum}

\begin{document}
    \lettrine[depth=1]% Add 1 line
    {L}{orem ipsum} dolor sit amet, consectetuer adipiscing elit. \lipsum[2]
\bigskip

        \lettrine[depth=2]% Add 2 lines
    {L}{orem ipsum} dolor sit amet, consectetuer adipiscing elit. \lipsum[2]
 \bigskip

        \lettrine[depth=1, loversize=-0.1]%add 1 line and 10% decrease.
    {L}{orem ipsum} dolor sit amet, consectetuer adipiscing elit. \lipsum[2]
\bigskip

        \lettrine[depth=1, loversize=-0.1, lraise=0.1]%add 1 line, 10% decrease and Go up 10%.
    {L}{orem ipsum} dolor sit amet, consectetuer adipiscing elit. \lipsum[2]
\end{document}

Output: