Taskbar = bar at the top of the screen, with time, shortcuts, battery indicator etc…
On my first note 1 still using 2.1.2 I can still have the taskbar always visible over any app. Which is awesome.
On my second note 1 upgraded to 2.2.1 it is no longer possible. Did I miss something? How to enable this again? Instant regret.
This is one of the reason I regret upgrading from 2.1.2 to 2.2.1 :-/
Go into Apps and open the Navigation Ball app. You can choose various functions for the five buttons. For one of them, choose the Full Screen function. Then exit the app, making sure, of course, that the Navigation Ball is enabled.
Then, when you are using a different app, just activate the Navigation Ball and tap on the Full Screen button. If you are already in Full Screen mode, then it should exit you from it and the task bar will become visible. Tap the button again to go back to Full Screen mode.
This seems to work well for most apps, but there are a few that it may not work for. It also doesn’t work for system apps like the native NeoReader. For system apps, you need to go into their settings to enable the task bar to be visible.
Your solution works brillantly for my note 1 on 2.2.2, thanks a lot!! 
However it does not work for my Note 2. The “full screen” option for the navigation ball is nowhere to be found unfortunately. The issue happens on many apps I use (youtube, slack, VLC…) where I still did not find a way to see the taskbar while using the app.
I found a fix for this and for more issues, check this: