I set up a secondary profile in Google Analytics a long time ago for a site for a store my wife owned. She has long since sold the store and neither of us has access to the site. I am not given the option to delete the "profile" anywhere in the Analytics dashboard. How can I get rid of it without removing my entire Google Analytics account?
Pelms is correct, you cannot currently delete a profile under certain cirumstances, Google support recommend requesting manual removal, however we never heard back from them. I'm afraid you are stuck with your extra profile.
Google Analytics organizes individual sites as profiles, and groups those profiles within "accounts".
Within your Google Analytics overview you should see all the accounts you manage. Click "edit" on the right for the account you wish to delete. Then within that next page you should see an option to delete the account. That will just delete the website account and not your entire user account.
If you wish to just delete one profile within an website account, first open that account by clicking its name and you can see a list of the profiles within that website account. There you can delete any profile using the delete link at the right.
I don't think you can currently if you're not an admin user - see this thread on the support forum.
One poster managed by emailing Google Analytics support and requesting manual removal but where he found the email address I've no idea. If you had/have an Adwords account you might have some luck here though.