Currently, the Stack Exchange logo in the top bar (on all sites except Stack Overflow, which has its own unique top bar) is a simple link to https://stackexchange.com/. I do not find it useful in any way, just annoying. I usually click it out of habit, and surely many new users click it without knowing where it will bring them.
To improve this, I suggest that clicking it will open a dropdown menu same as the icons to the right:
The menu will consist of:
- No more throwing us away from the current page when we never intended to.
- People will actually see where they are going to.
- Links to useful pages in stackexchange.com
- Consistency with the behavior of other icons in the top bar.
† - network profile link can be hidden for visitors who are not logged in.