The default character limitation for the native Joomla! Search Module is 20 characters.  Increasing this value requires a bit of Joomla! hacking, and although there are a number of resources for accomplishing this, I could not find any that pertained to the current (or even recent) Joomla 1.5 release.

NOTE:  You should not remove the maxlength entirely.  As long as search times are non-critical to you, and your database is not enormous, making it 50 or 100 characters should make for a more appealing search interface in most cases.

In order to change the search character limitation to 45, make the following changes:

Note: The ” – -”  means delete this, and the “++” means to add this back

/modules/mod_search.php

-- $maxlength		 = $width > 20 ? $width : 20;
++ $maxlength		 = $width > 45 ? $width : 45;

/components/com_search/views/search/tmpl/default_form.php

--

++

/administrator/components/com_search/helpers/search.php

--    // limit searchword to 20 characters
--    		if ( JString::strlen( $searchword ) > 20 ) {
--   			$searchword 	= JString::substr( $searchword, 0, 20 );
++    // limit searchword to 45 characters
++    		if ( JString::strlen( $searchword ) > 45 ) {
++   			$searchword 	= JString::substr( $searchword, 0, 44 );

/language/en-GB/en-GB.com_search.ini

-- SEARCH_MESSAGE=Search term must be a minimum of 3 characters and a maximum of 20 characters.
++ SEARCH_MESSAGE=Search term must be a minimum of 3 characters and a maximum of 45 characters.

There you have it, enjoy!

Thanks to the following:
leeburstroghm – for pointing out that this change requires mods to the admin component in order to function properly on the front-end.

m2d^ – for a helpful post that accomplished this update in an older version of Joomla! / the Joomla! Search Component. It’s close, but just distinct enough to make things tricky.