I've never used Typeahead before, but I checked their docs and their demo-page. And it sure was kind of tricky.
Anyhow, got it to work with this simple script that you can paste in your browser console, which forces you Typeahead searchbox to open, and stay open while you poke around in the element inspector:
$('.typeahead').focus().typeahead('val', '').focus().typeahead('val', 'what ever value you want to test');
At least in chrome, the suggestion box even stays open on right click > inspect element.
Update; the OP asked for an more extensive explanation:
The code above is jQuery and takes a selector – the element that contains the Typeahead. A screenshot:
The selector in this example is specific for the demo-pages for the Typeahead library. You might need to tweak it depending on how you initialize your own Typeahead.