We'll create fresh WordPress site with Autocomplete For Relevanssi installed. You have 20 minutes to test the plugin after that site we'll be deleted.
This plugin is meant to add the word autocomplete in search input field by using the index stored in Relevanssi database. So you have to install first Relevanssi (free or premium version) to have it to work.
The JS script used to accomplish the autocompletion work has been developed by Lea Verou and is called Awesomplete. It is very simple to set up, lightweight and accessible.