That is not an option, a stacked baloon must have some type of border, try using a note.
Well, I was able to figure out how to do it, but It doesn't work very well. You can stack the balloons and leaved them framed to start, then if you click on the balloon that you would like to remove the frame, under more properties - change the border to "none".
The problem: Chances are the non-framed balloon will overlap with the adjacent balloons.
I tried ad nauseum to find the thread where this was already discussed. I could not. Here is my screenshot again. This is the same method you showed, I use it often.
Also, and probably more importantly, SW2010 supposedly has a function to do this (I'm still running '09 for production). From the SW2010 What's New PDF:
Use Numbering from Specified BOMs
When you insert balloons and stacked balloons into an assembly, you can set the balloons
to follow the item numbering of a selected BOM in the assembly under Balloon text
source in the Balloon PropertyManager. You can then import the balloons into a drawing.
See SolidWorks Help: Balloon PropertyManager.
You can set a quantity value in the Balloon PropertyManager that updates parametrically.
In previous versions you had to add text outside of the balloon, then group the text with
the balloon. The text did not update parametrically.
I stand corrected, Thanks Charles, never used it that way.
Changing the frame to NONE in More Properties worked for me in SW2014. I needed the ability to create parametric notes to potentially changing BOM numbers. We are using Driveworks to automate output drawings and when the BOM changes by a part getting removed, notes referencing that line item need to change numbers. My company wont let me keep BOM numbers fixed. I can now use a set of hidden bubble notes without borders to copy into my notes and have notes that update with the BOM.