In org mode, this inline latex dollar sign fails to render:
but this has no problem rendering:
\[ \$ \]
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Although Org mode allows LaTeX math fragments to be inserted, there are some limitations. This is one of them. The way to get inline math fragments in such cases is to use the
\( ... \) form that LaTeX allows, e.g.
\( \$ \)
in this case. See the LaTeX fragments section of the manual, which states:
LaTeX fragments do not need any special marking at all. The following snippets are identified as LaTeX source code: ... Text within the usual LaTeX math delimiters. To avoid conflicts with currency specifications, single ‘$’ characters are only recognized as math delimiters if the enclosed text contains at most two line breaks, is directly attached to the ‘$’ characters with no whitespace in between, and if the closing ‘$’ is followed by whitespace, punctuation or a dash. For the other delimiters, there is no such restriction, so when in doubt, use ‘(...)’ as inline math delimiters.
Note that your attempt
$\$$ satisfies all the rules - but it still does not work, so that's a bug in the documentation.