If you were to change the stocks without being logged in, once you navigate away from the site, the symbols would be lost, and I felt users might be annoyed by that. Registering literally takes seconds too.

Yes, but that is seconds users do not want to take. I'd advice letting them edit it, but have a small notification at the top letting them know that it will be lost without an account. Cookies are unreliable but they may help.

That's a very good idea and I'll definitely consider implementing that soon. Thanks.

You can temporarily store the user's list in a cookie if they're not logged in.