Add JavaScript to a page without access to core files?
If the developer has granted your user role access to the appropriate input formats and and has it configured to allow JS then yes, you can.

You may see a select box under one or more of the fields that allows you to select a filter such as "filtered HTML", "Full HTML" etc..

This is however normally locked down to prevent content editors from breaking a site, so I would be surprised if your limited role had access to unfiltered HTML/JS input.

A quick way to test if it works would be to add &lt;script&gt;alert('test');&lt;/script&gt; in the box and save/refresh the page.