I forgot about the meaning "greater". I think Arabic does not dinstinguish between comparative "greater" and superlative "the greatest".
You are almost correct. I thank you for pointing that out.
There's another form of noun in Arabic language termed Seeghah mubalaghah.
For example, you are correct that if Allah is great the arabic equivalent is Allahu Kabeer. but since the term is Allahu Akbar, the proper translation is Allah is Greater.
To say Allah is Greatest, the word kabeer is used. In fact, one of the attributes of Allah is Al Kabeer. (The Greatest) with the prefix "Al" denoting a definite attribute.
Seegha mubalaghah is not only used to give the superlative attribute but also gives Allah the attribute of ALWAYS having that quality.
For example, the word Raheem is a seeghah mubaalaghah of raahim. (assuming that you know Arabic language, this is the verbal noun [ism faa'il] of the verb rahima [to be merciful].
Now Allah is also raahim [merciful] but to use Raheem [double vowel shows prolongation of sound] would denote he's not only the Most Merciful but this attribute is ALWAYS with him.
Unlike us, humans, we may be "raahim" or "raahimah [female], but we are not ALWAYS merciful to others.
In a precis, the Akbar is used to note that Allah is greater than anything else complemented by another attribute of al kabeer.
I hope that clarifies the matter a bit.
You may want to check Antoine Dahdah collection of Arabic grammar and syntax. His are the best ones out there.