Custom jQuery plugin not working

First, your link 403’s for me. I assume you’re using the code I found here:

https://github.com/trentrichardson/jQuery-Timepicker-Addon/blob/master/dist/jquery-ui-timepicker-addon.js

Now, the $ thing is not a problem, that JS code library uses a proper $ wrapper, so it will work fine without modification.

The actual problem is a simple misspelling. Look at this line of code:

wp_enqueue_script('timepicker', get_stylesheet_directory_uri() . '/js/timepicker/jquery-ui-timepicker-addon.js', array('jquery', 'jquery-ui-core', 'jquery-ui-slider', 'jqueryui-datepicker', ), '1.4.3' )

That bit where you have jqueryui-datepicker should actually be jquery-ui-datepicker. Note the additional hyphen.

Look in wp-includes/script-loader.php, around line 163 or so, and you’ll find where all the default scripts are named.