You have to change this in your template overrides to make the input box bigger, but you also have to change it in a core component.

The core for the form is here:

components –> com_search –> views –> search –> tmpl –>default_form.php

Copy that and put it in your html folder for your theme:

html –> com_search –> search –> default_form.php

Do this for  mod_search as well

path will be:

html –> mod_search –>default.php

That takes care of the form, but the real limitation is in the joomla core:

administrator –>components –> com_search –> helpers –>  search.php

find these lines:  (starts around line 58)

and change limits  (20 –>50 and 19 –> 49)

function limitSearchWord(&$searchword)
 {
 $restriction = false;

 // limit searchword to 20 characters
 if ( JString::strlen( $searchword ) > 20 ) {
 $searchword     = JString::substr( $searchword, 0, 19 );
 $restriction     = true;
 }

 // searchword must contain a minimum of 3 characters
 if ( $searchword && JString::strlen( $searchword ) < 3 ) {
 $searchword     = '';
 $restriction     = true;
 }

 return $restriction;
 }

You'll have to re-do this whenever you update to a new version of Joomla!