It's because you try to add again and again the same element (instead of a new one). Let's create a new one for each click.
Modified code from your JSFiddle example:
$("#btnAddWord").on('click', function(){
var inputGroup = document.createElement('div');
inputGroup.setAttribute('class','input-group');
inputGroup.innerHTML = '...here your HTML...';
$('div.input-group:last').after( inputGroup );
});