I decided to use a heredoc and output to a file to determine what effect, if any, my first call to mysql_real_escape_string() was having. I used the following code -- the php variable "theTextWithManyQuotes" was read from user input, it was a text string such as "Isn't O'Malley's parents' children's "choices" atypical"
$theTextWithManyQuotes = mysql_real_escape_string($_POST['userInput']);
$html = <<<HEREDOC
<!DOCTYPE html>
<body>
<textarea readonly name="adPreviewText" id="adPreviewText" rows="4" cols="60"
style="border: none; border-style: none">$theTextWithManyQuotes</textarea>
/body>
</html>
HEREDOC;
file_put_contents("testfileonly", $html);
I then dumped the $html variable to a file by way of file_put_contents() and opened the "testfileonly" file -- and the backslashes were in fact present in the text.
My surmise is that when I build a mysql query string and it contains escaped text strings that were escaped by a single call to mysql_real_escape_string(), the database somehow 'hides' the backslashes so that they're not visible in phpMyAdmin when looking at the database record.
The fact that my heredoc's contents, when output to a file, shows that the backslashes are present, it proved to me that only a single call to mysql_real_escape_string() was required, and the mysql database is somehow not showing (or stripping?) the backslashes in the database records. When the data is read back out of the database, no call to stripslashes() is required, the backslashes are not present in the text strings when read back out from the database.