A great American consulting firm official
What does this "great" modify? Could it modify "consulting firm?"
I think it can only modify "official". If it can modify "firm", then it should be
A great American consulting firm's official
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It can be modifying consulting firm or it can be modifying official.
Consulting firm is a noun which is restricting the sort of official, another noun, that we are discussing.
American is describing one of the nouns. Although the same word could be used as a noun in a different phrasing.
Great is describing one of the nouns.
If you wanted to be truly unambiguous...
That great American is an official at a consulting firm.
The American [person] is [acting as] an official at a consulting firm.
He is an official at a great American consulting firm.
The consulting firm is both great and American, and he is an official there.
He is a great American official at that consulting firm.
That official is both great and [an] American [person], and he officiates at that consulting firm.
This sort of ambiguity is not uncommon:
For sale : Queen Anne table by nice lady. With nice legs and big drawers.
Here are two more possibilities.
There are theme parks named "Great America". It is possible that
A Great American consulting firm official
is an official who works for a consulting firm that does work for the Great America company. Or the Great America company could own a consulting firm, and the official could work for that consulting firm.