Here is a sentence I wrote for an exercise:

Oğlanın on iki beysbolu ve üç beysbol sopasını var.

The dictionary here at Turkish Class seems to tell me that "beysbol sopası" is what my lesson book calls a Type II possessive construction. (Sorry, I have forgotten the term I learned here. Help me!) From the dictionary: beysbol sopası baseball bat. 

In the above sentence, I want to say the boy "has" a baseball bat, so I need to use the possessive along with "var," yes? So, it seems I would need to add the buffer letter "n" and another "i" to "beysbol sopası." Here is the catch: I have been told this is wrong -- that it would just be "beysbol sopası." I am all confused now. Can someone explain, lütfen? Teşekkürler!

Right form of your sentence in Turkish.

Oğlanın, on iki beysbol topu, üç beysbol sopası var.

I think I have found my explanation, in Lewis Thomas´s "Elementary Turkish":

"In the case of a noun whose absolute form ends in a vowel, note the ´disappearance´ of the third person possessive suffix when its possessive form becomes a member of a possessive complex."

"onun yatak odası"  --  "his bedroom"

so

onun beysbol sopası  --  his baseball bat

Is this what is happening?

There is no problem with onun. But what happens if you use senin?

o zaman "senin yatak odan" oluyor

Hmm. Yes, I understand what you are both saying. But I still don´t think I understand why "beysbol sopası" doesn´t take anymore suffixes. Is it this?:

"The second member of the Type II complex already has a possessive suffix. No word may bear more than one possessive suffix. Hence the -si does double duty, serving (a) as possessive suffix of the Type II possessive construction and (b) as possessive suffix of the possessive complex."

So, does it matter about whether the absolute form ends in a vowel, or am I not adding more possessive suffixes simply because a Type II construction never takes more possessive suffixes?

And so, when the absolute form ends in a vowel and we are not in third person, the -si disappears in favor of regular possessive endings. Thus, there is still only one possessive suffix in the complex. But what happens if the absolute form doesn´t end in a vowel? Aaargh!

A short explanation:

"şarkı sözü"

şarkı sözü : undefined noun phrase

şarkının sözü: defined noun pharase

onun şarkısının sözü: chained noun phrase

senin şarkı sözün: chained noun phrase

blue: modifier

red: modified

You don´t use two possesives because there is always one modifier and one modified.

 Quoting uzma 321