Edit in JSFiddle

var i=1,
    container=$('<div>'),
    addToList=$('#addToList');

addToList.after(container);
container.append('<select id="dropDown" size="10" style="width:200px">');

addToList.click(function(){
    $("#dropDown").append($('<option />').text('Item '+i++));    
});   
<input type="button" id="addToList" value="Add to List" /><br/>