I tried to edit paradd0x's answer above to include the idea of using Ctrl+Alt+Home as that more exactly answers the question, but it was rejected by the mods. Kind of annoying, but whatever.
So what I think is the easiest thing to do here is to:
- Uncheck Display the connection bar when I use the full screen as already mentioned, which prevents the connection bar from popping out on you when you mouse to the top of the screen
- And then when you want to see the connection bar press Ctrl+Alt+Home which is precisely the key combination assigned to making it show
This keeps the connection bar out of the way until you specifically summon it. And once you've summoned it, simply clicking anywhere else will make it go away completely until you summon it again.
This seems to be exactly what you're wanting, no? I use this myself (in conjunction with the below).
An alternative to consider:
I use Chrome, which has the same "problem" with the tabs. I'm also a long-time Winamp user and use it in Windowshade mode at the top of my screen, with Always on top set to true. When Chrome isn't maximized it leaves just enough room for Winamp to hover above the tops of the tabs. This combined with the fact that maximizing any web browser on a 24" monitor (all my monitors are 24") is a waste of horizontal space, means that I literally never maximize Chrome.
As an aside, I use WinSplit Revolution (main site is now dead, search for it on Google and you can find it at one of the download sites) which is a (free) utility that places/sizes the focused program wherever you want. For instance, pressing Ctrl+Alt+Numpad5 will position Chrome (or whatever's focused) in the center of the screen at 66% width and 100% height, which is my preferred browser size on a 24" monitor. Additional presses of the same keys cycle through different placements/sizing, one of which is 100% width, 100% height, but not maximized -- this would allow you to use your browser "full screen" if that's how you prefer, but still have the tabs pushed down away from the top of the screen. WinSplit is the single most used/useful utility I've found in my 15 years of working with computers.
Anyway... where this applies to Remote Desktop is Winamp will actually show over the top of even a full screen Remote Desktop session. When I want to switch back to my local machine, I simply click Winamp once and the taskbar from my local machine appears -- voilà!. I mainly RDP into my workstation, so I prefer not to minimize the remote session. Running local programs at less-than-full screen width leaves portions of the remote session visible in the background (not to mention completely visible on my second monitor) allowing me to keep tabs on incoming email, new IMs that pop up, etc. I'm so used to the convenience of this that I run Winamp even when I don't have it playing any music.