$re = '/\[\[(\{\{.+?\}\}|[^^{}]+)\^(.+?)\]\]/m';
$str = 'Example String: [[{{hello^test}}|{{goodbye}}]]
Expected Match: False because i want to only match the ^ if it\'s part of the outside part of the string surrounded by [[ ]] not if it\'s inside the {{ }}
Example String: [[{{hello^test}}^{{goodbye}}]]
Expected Match: True because both sides of the ^ inside [[ ]] match, ignoring what is inside the {{ }}
Example String: [[{{hello^test}}^{{goodbye^test}}]]
Expected Match: True because again the ^ is between the two parts in {{ }} and needs to ignore what is inside {{ }}
Example String: [[{{hello^test}}|{{goodbye^test}}]]
Expected Match: False because there is no ^ inside the [[ ]] only inside the {{ }} which should be ignored
[[{{6^-2}}x{{6^3}}|{{6^-2}}]]
[[{{6^-2}}x{{6^3}}^{{6^-2}}]]
';
preg_match_all($re, $str, $matches, PREG_SET_ORDER, 0);
// Print the entire match result
var_dump($matches);
Please keep in mind that these code samples are automatically generated and are not guaranteed to work. If you find any syntax errors, feel free to submit a bug report. For a full regex reference for PHP, please visit: http://php.net/manual/en/ref.pcre.php