Do you have pictures?
What is the material and structure of the floor?
What is above the ceiling?

here is a space with a similar ceiling height... we put a skylight and removed the suspended ceiling.

Yes, I would like to see pics. You must have a cement floor? The issue with digging down is that you may have to undepin the existing conc. foundation in order to reinforce the existing support for the exterior walls that they wont overturn when you remove, regrade and repour the existing slab and thermal break.--you'll have to furr out and insulate any exposed conc. to meet energy code on the inside face, if you cannot achieve energy code standards with rigid insulation applied to the outside of the foundation. The other option is to do something with modifying the roof structure, such as removing the ceiling joists (if stick framed) or removing the roof in it's entirety, if trussed now, and retrussing the roof to achieve add'l height. Both of these options, even in terms of partial gains, are problematic and expensive, and should only be undertaken after a consult with a qualified ("we've done this type of project before") architect/engineering team. This is specialized work, and not a typical type of remodel project. Best wishes with your endeavors.

- Why would an open floor plan require ceilings taller than 7'6"?
- No, excavating would probably not be worth increasing the ceiling height since 7'6" isn't functionally prohibitive.
- If the space above the ceiling is an attic then it would make better sense (cents) to drop the ceiling drywall to create an exposed joist ceiling. The perceived ceiling ht. would be the width of the CJs. Another option would be to create cathedral ceilings up to the height of the attic collar ties (keeping the existing CJs or every other CJ in situ as an option)

Its much easier to raise a roof then dig down...
7'6 is very short, especially for an open floor plan.
The wider the room, the lower the ceiling will feel, its a proportional ratio.

7'6 is 7'6, which is only acceptable for passageways, attics, and closets.

7'8":

I don't think I would spend the money on excavating for 6" to make 8' ceiling . I guess if the other ceilings are alot higher than 8' then you might have to. I find sometimes when I encounter what may be a problem it becomes something interesting in a design so if you send a little more info and some pictures of the space we might be able to be more helpful.

Just to let you know the ceiling in this picture was only 7'