As of Bootstrap 3.x, there's an example of this in the docs here: http://getbootstrap.com/components/#input-groups-buttons-dropdowns
<div class="input-group">
<input type="text" class="form-control" aria-label="...">
<div class="input-group-btn">
<button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" aria-expanded="false">Action <span class="caret"></span></button>
<ul class="dropdown-menu dropdown-menu-right" role="menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div><!-- /btn-group -->
</div><!-- /input-group -->