The short answer to your question is no. As Dominik said, the 'primary logo as a link' method should return users to the home page. Every functional website you can imagine does this.
However, if you want to put a sub-menu ('notification', 'hot topics', 'new' etc.) in the upper left near the logo, that will work just fine so long as it is a separate element from the logo itself.
One exception to this is the app Vine. In Vine, you tap anywhere on the top navigation bar and it pulls a dropdown menu down. This is different, however, because the logo is located in the center, not the corner. Also it's an ultra-minimal video iOS app, not a 'regular' website.
If you want to be innovative, follow VARAK's advice and make hi-fi mockups and ask friends to navigate and iterate from there. Otherwise stick to what people know— the logo will go to the home page, a dropdown nearby can show secondary info, and a full site nav should remain on a nav bar to the upper right or a nav column on the left or right side of the screen.