"I want to be famous!"
Translation:Ben meşhur olmak istiyorum!
'Olmak' is technically a noun. But it can't take an accusative, seemingly. But if there is a need for accusative 'olmayı' would be used(In fact its bare form is 'olma'). But it sounds a little bit odd here. But don't be surprised if you hear someone using it. But here are some examples about 'olmayı' and 'olmak' (using also 'ünlü'):
Ünlü olmayı sen istedin. (You wanted to be famous.)
Ünlü olmak o kadar kolay değildir. (Being famous is not that easy.)
Ünlü olmak için çok bekledim. (I waited for being famous so long.)
Ünlü olmayı çok bekledim. (I waited so long to be famous.)
In your first and last example, olmak was the object of a verb and therefore became accusative olmayı. In the other two, it was not the object of a verb and therefore remained olmak.
However in the original sentence, "Meşhur olmak istiyorum," it is the object of istiyorum, but it does not become accusative!
In summary...I'm still confused.
Okay then let me say if a word will be put between 'olmak' and conjugated 'istemek' you are supposed to use 'olmayı' and istemek. Because indefinite direct obejct(non-accusative) can't take word between the verb and itself.
Ünlü olmak istiyorum.
Ünlü olmayı çok istiyorum.
That case is similar to gerund and infinitive distinction in English. Anyway, here are some other sentences with -mek or -me:
Evde yemek yemeyi seviyor. (He/She likes eating at home.)
Kitap okumaktan hoşlanıyorum. (I like/enjoy reading books.)
Okulda kalmakta ısrar ettin. (You insisted on staying at school.)
Sınavları geçmeye çalışıyorum. (I am trying to pass the exams.)
I think like the noun cases, this is something that shold be memorized.
It seems like the gerund/infinitive question is more about phonetic flow than the grammar of the thing. Would you say that's right?
We've got "Ünlü olmak istiyorum," and "Ünlü olmayı sen istedin." What about, "Ünlü olmak istedin"? Without the sen, olmak seems better phonetically, so I'm guessing that's right.
It all depends on if the infinitive comes directly before the verb or not. If it is directly before the verb, it remains the same. If anything whatsoever come in between, you must add the accusative case. This actually stands true for all objects in Turkish.
"Ben kitap okurum." "Kitabı ben okurum." are both fine but "Kitap ben okurum." is simply wrong. :)