It does on gnome shell and cinnamon (linux mint). Why not in unity?
In Unity the "top panel" is a menu bar only, when an application is running in Unity you get an arrow on the left side of the icon (and an additional arrow on the right when the application is focused)
To add to Jorge's answer, multiple instances of a program show up as multiple white dots to the left of the Launcher icon (up to three I believe). Single white triangles on the left indicate active processes.
Clicking on the button on the unity bar displays all the windows or fades the desktop and brings up small versions of the windows that you can click on to select the one you want.
In Oneiric, the left side indicators (dots or triangles) turn blue when the application wants attention. (In Natty, a blue triangle appeared in the upper left on the Ubuntu button, but that button is now on the top of the Launcher in 11.10.)
Also, depending on your Backlight settings for Unity, the application that wants attention gets backlit with edge illumination. Below, you should see the Synergy and GEdit icons are not backlit, while the Synaptic and the Gimp icons are. The GIMP icon also has the subtle edge illumination effect. (And The Gimp has a bug where it always requests attention when starting up.)
Use the Compiz Configuration Settings Manager to change the backlight mode. Mine was defaulted to "Backlight Always On" - so all the icons were backlit (as in the earlier screenshots). Changing it to "Backlight and Edge Illumination Toggles" now has the active windows backlit and the ones wanting attention edge-illuminated.
See also this question on how to bring the demanding application to focus. and this one on auto-raising chat windows when demanding attention
The Launcher should provide as much functionality as seeing Chrome in the top panel. You can also make extensive quicklists that show customizable menus when right-clicking on the Launcher icons.
See this excellent question: What custom launchers and quicklists are available, and in your case in particular: Chrome quicklist