Yes, We are talking about Office 365 and Sharepoint Online & suprisingly this applies to Plan P1/E1/E2/E3. Couldn't believe it huh??
First-off, you shoudl know using plan P1 you could still create a sharepoint site for anonymous users. Now, we have a solution for granting additional privileges to anonymous users for specific list/document libraries. It's free & you can download a Sandbox Solution (GAU) here. Just upload the WSP to your sharepoint website that has anonymous access & activate it.
Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.

|
View items in lists, documents in document libraries, and view Web discussion comments, allow users to open a Web site, list, or folder to access items inside that container & View the source of documents with server-side file handlers. |
Add List Items |
Add items to lists, add documents to document libraries, and add Web discussion comments. |
Browse Item |
Enumerate files and folders in a Web site using Microsoft Office SharePoint Designer 2007 and WebDAV interfaces & View information about users of the Web site. |