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How would I make only specified paragraphs have a little extra space above and/or below them compared to the default setting (which seems to be no--or very little--extra space between paragraphs)? An entire newline is too much space (i.e., \\* is too much space since it adds an entire blank line).

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I would define a macro and manually insert it above/below the specific paragraphs you want adjusted in this way:

enter image description here

\documentclass{article}
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\newcommand{\extraparskip}{\bigskip}
\begin{document}
\lipsum[1-2]

\extraparskip

\lipsum[3]

\extraparskip

\lipsum[4-5]
\end{document}

The advantage of such an approach is that you can adjust the macro in the preamble and obtain a globally adjusted effect. I've used \bigskip above, however, you might be interested in something more subtle like \medskip or \smallskip.

You could also define an environment and encompass the specific paragraph(s) within it. However, this groups the paragraph contents (making changes within local), which could affect other things.

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