This is straight out of the Poptropica FAQ website:
Q: I forgot my user name and/or password how can I retrieve it?
A: Unfortunately, if you've forgotten your username, there is no way to retrieve it. We do not store any private information about our users. We recommend you create a new avatar and write your username down on a piece of paper and keep it in a safe place!
Players can register a parent email address with Poptropica so that they can reset a child's password if it is forgotten. Your child must have input your email address through his/her player account and had the address verified specifically for this purpose before the "forgot password" function will work. Please note that parents cannot change a player password through their parent account. To reset the password, begin by entering the username on the Returning Player login page, without entering any password, and then click the "OK" button. Poptropica will display a "forgot password" link. Click that "forgot password" link, and then click the "OK" button. Poptropica will send an email to the parent with instructions for creating a new password for the child's account.
I'm sorry to tell you, but unless you remember really hard or you put in an email, I don't think you can get it back. I know how frustrating that must be when you've completed so many islands and have a paid membership.