Learn Arabic (Modern Standard): أين مدخل مكتب الرئيس؟

Just like a two-word idafa, you stack the nouns together. We have مدخل (entrance), مكتب (office), and الرئيس (the boss). Literally, it reads as 'entrance of office of the boss'. You can chain as many nouns as you need this way to show possession!

Why don't the words for 'entrance' or 'office' have the definite article الـ (the) on them?

In an idafa chain, only the very last noun is allowed to have الـ. The earlier nouns, مدخل and مكتب, automatically become definite because they are 'owned' by the final definite noun الرئيس. Adding الـ to the beginning or middle of the chain breaks the grammar rules.

If I want to read the grammatical case endings aloud, what would they be for the idafa chain?

The first word in the chain, مدخل, is the subject here so it takes a 'u' sound (Damma). Every noun that comes after it in the chain acts as a possessor (genitive case), so both مكتب and الرئيس take an 'i' sound (Kasra). Read fully aloud, it sounds like 'madkhal-u maktab-i r-ra'iis-i'.

I usually see الرئيس translated as 'the president'. Can it also mean 'the boss'?

Yes! While الرئيس is the formal word for a president of a country, its core meaning is 'head' or 'chief'. In a normal workplace setting, it is perfectly standard to use it for the boss, the director, or the head of a department.