Since this question was posted almost a week back, I decided to post the answer here instead of commenting over there (Also, it would be too long of a comment too!). Unless you are an Arabic grammar(Nahwu – نحو)/morphology(Sarf – صرف) geek like me, you may not find this post interesting at all :) . Please note also that some of the english terms that I am using are terms that I made up in an attempt to translate the arabic term as accurately as possible. I prefer using the Arabic terms. And if you are not too familiar with English and Arabic grammar, you may find this hard to understand. You may also want to increase the font size to see the Arabic words. Let's begin!

As answered on Zacharia 's blog, Subhana-Allahi is actually two words:

1 – Subhaana سُبْحَانَ: It literally means to exalt and make something higher than and above all things that lower its status

2 – Allahi اللَّهِ – Subhanahu wa ta3aala

Subhaana is an infinitive form (in Arabic terminology known as the مَصْدَرْ) of the verb sa-ba-ha (سَبَحَ). An infinitive is part of a branch of nouns called (Verb-like nouns – الأَسْماء المُشَبَّهَة بالأَفعال). These nouns are similar to verbs in that they indicate an action which has been performed. This class actually consists of 9 different types of nouns:

a) The infinitve – المَصْدَر
b) The Subject Noun – اسْمُ الفاعِل
c) The Object Noun – اسْمُ المَفْعول
d) The Subject Noun-like Adjective – الصِّفَة المُشَبَّهَة باسْمِ الفاعِل
e) Exaggeration Forms – صِيَغُ المُبالغَة
f) Preference Nouns – اسْمُ التَّفْضيل
g) Positional Nouns – اسْمُ المَكان
h) Durational Nouns – اسْمُ الزَّمان
i) Nouns which describe machines – اسْم الاّلَة

Anyway, an infinitive form indicates an action without giving any reference to time, and it consists of the letters which make up its verb. Example:

Da-ra-ba ضَرَبَ (hit) —> Darban ضَرباً (act of hitting)

Qa-a-ta-la قاتَلَ (fought) —> Qitaalan قِتالاَ (fighting)

For those of you who are familiar with the sentence structure of the Arabic language, there are two types of sentences:

a) Verbal Sentences (جُمَلْ فِعْلِيَّة) which begin with a verb

b) Nounal Sentences (جُمَل اسمِيَّة) which begin with a noun

Subhaana is not a verb; it is a noun (an infinitive – مَصدَر) as stated above, and it has the form (وَزْن) "Fu3laan – فُعْلان". Another similar example would be gha-fa-ra غَفَرَ (forgave) —> Ghufraan غُفْران (Forgiveness). Since subhaana is a noun, then it should have the grammatical state of Raf3 (رَفْع) since it is the beginning of the Nounal sentence (called مُبْتَدَأ); i.e. it should have a Dammah (ضَمَّة) on the last letter making it subhaanu (َسُبْحانُ). This is obviously not the case. Hence the sentece is not a nounal sentence. The sentence must be verbal then. Where is the verb?? It is not mentioned (It must have been omitted – مَحذوف).

The phrase "Subhaana Allahi" is well known and has been used for a very long time, and as such, grammaticians know what this phrase means and what the omitted verb ought to be. The say that original phrase without omitting the verb should read أُسَبِّحُ سُبْحانَ اللهِ which can be translatied as "I praise Allah exalting Him above all attributes that do not suit His majesty". So the verb أُسَبِّحُ was omitted. The verb sa-ba-ha has many meanings but the most accurate one here is "to praise".

Obviously, the noun Subhaana in the state of Nasb نَصْب because of the Fatha فَتْحَة on the end. In a verbal sentence, nouns which are mansuub مَنْصوب are part of the family of Mafaaa3eel مَفاعيل . Without mentioning them all, I'll can say for sure that Subhaana is not an object مَفْعول بِهِ because subhaana is not the thing that that action of praise was done to. Instead, subhaana describes the type of praise (in this case it is exalting Allah above all derogatory attributes).

Coming to the answer (you are thinking: "OMG, this guy could have just said it from the beginning!"), an infinitve that indicates the number of times an action has been performed, or the type of action, or the repetitivity of the action, and which has the same form of the verb is called the absolute object مَفعول مُطْلَق . It is always Mansuub مَنْصوب and its verb can be omitted (this technique of omission is used in speeches as part of Arabic elequoence and conciseness).

So the answer is:

Subhaana is a مَفْعول مُطْلق. And it is obviously a mudaaf مُضاف like Zacharia said, and Allahi is Mudaaf Ilayh مُضاف إلَيْه. But just as a note, a mudaaf can have any harakah حَرَكَة on it, and by saying a word is a mudaaf, you are not specifiying its grammatical place in the sentence. Mudaaf Ilayh on the other hand fully speficies a nouns place in the sentences, and it always comes as Majruur مَجْرور.

(Note in another book I read, it says that Subhaana is نائِب مَفْعول مُطْلَق (something that substitues for the Absolute object)).