I am not trying to make it correct or anything else. It´s what it is and I only tried to describe it and while doing it, I do not see any incorrectness in it.

I bet you also say "xx yaşındayım" at the times you do not make yourself force to use "xx yaşımdayım" form.

Thank you, Henry, it didn´t escape your sharp eye. I wonder how it didn´t ring any bell in me. Maybe I do the same mistake all the time. I must confess I mix gitmek and gelmek also.

By the way, in Swedish teaching and learning are expressed with the same verb, except that when used for learning a reflexive pronoun belongs to the package. Thus, lära ´to teach´ and lära sig ´to teach oneself > to learn´. I like the thought.

It is the same in Polish:

to teach - uczyć (alternative: nauczać )

to learn - uczyć się (literally it could be translated as "to teach yourself"; although I wouldn´t be sure if word "się" in Polish always makes verb reflexive).

Also Cahit Sıtkı disagrees with you on this..

Or you think it should be "Otuz Beş Yaşı"?

Here is the poem

OTUZ BEŞ YAŞ

Yaş otuz beş*; yolun yarısı eder.
Dante gibi ortasındayız ömrün.
Delikanlı çağımızdaki cevher,
Yalvarmak, yakarmak nafile bugün,
Gözünün yaşına bakmadan gider.
Şakaklarıma kar mı yağdı ne var?
Benim mi Allahım bu çizgili yüz?
Ya gözler altındaki mor halkalar?
Neden böyle düşman görünürsünüz,
Yıllar yılı dost bildiğim aynalar?
Zamanla nasıl değişiyor insan;
Hangi resmime baksam ben değilim.
Nerde o günler, o şevk, o heyecan?
Bu güler yüzlü adam ben değilim;
Yalandır kaygısız olduğum yalan.
Hayal meyal şeylerden ilk aşkımız;
Hatırası bile yabancı gelir.
Hayata beraber başladığımız,
Dostlarla da yollar ayrıldı bir bir;
Gittikçe artıyor yalnızlığımız.
Gökyüzünün başka rengi de varmış;
Geç farkettim taşın sert olduğunu.
Su insanı boğar, ateş yakarmış;
Her doğan günün bir dert olduğunu,
İnsan bu yaşa** gelince anlarmış.
Ayva sarı nar kırmızı sonbahar;
Her yıl biraz daha benimsediğim.
Ne dönüp duruyor havada kuşlar?
Nerden çıktı bu cenaze? ölen kim?
Bu kaçıncı bahçe gördüm tarumar?
Neylersin ölüm herkesin başında.
Uyudun uyanamadın olacak.
Kimbilir nerde, nasıl, kaç yaşında?
Bir namazlık saltanatın olacak,

 Taht misali o musalla taşında. 

*There is no need for a detailed explanation as the title of the poem and the first line of it tell everthing:

Otuz Beş Yaş,

Yaş otuz beş...

What is age?

=>35

What is 35?

=>age

There is no need for poss. -ı

If there was,

** insan bu yaş-a gelince

should have been written as

insan bu yaş-ı-n-a gelince

As for the part

Uyudun uyanamadın olacak.
Kimbilir nerde, nasıl, kaç yaş-ın-da?
Bir namazlık saltanat-ın olacak,

 It is clear from the personel ending -n and the possessive suffix -ın that it is second person...

I admit it is a bit impolite but we can also say "yaş kaç?".. can´t we?

kaç yaş <=>yaş kaç

If it was with poss.-ı, it would indicate 3rd person in both version: 

kaç yaşı <=> yaşı kaç

Likewise

otuz beş yaş <=> yaş otuz beş

If it was with poss.-ı, it would indicate 3rd person in both version:

otuz beş yaşı <=> yaşı otuz beş

...

Yaş otuz beş = years are 35 (meaning I lived 35 years)

So what?

Here´s a final wrap up on my side.

yıl and yaş both mean year.

35 yaş = 35 years

35 yaşım = my 35 years

35 yaşımda = in my 35 years

35 yaşımdayım = I´m in my 35 years

35inci yaş = 35th year

35inci yaşımda  = in my 35th year

35inci yaşımdayım = I´m in my 35th year (i.e. I´m 35 years old)

35 yaşı = year of 35

35 yaşında = in the year of 35

35 yaşındayım = I´m in the year of 35 (i.e. I´m 35 years old)

I have tried to read carefully everything that natives have said in this thread. To sum up everything as I have understood it I noticed the following:

The problem about the structure 35 yaş|ı|n|da|yım was the adjective nature of Turkish numerals. In a compound they don’t demand a POSS in the next noun. For a while I was suspecting this because all adjectives as a matter of fact can be used as nouns. But it seems that numbers are especially sensitive adjectives: they hardly ever stand on their own, there has to be a noun after them (tane, kişi). The problem remains.

As the POSS in yaş|ı|n|da|yım can’t be motivated by the compound it has to be motivated by something else. I think there are two alternatives that have been mentioned in the course of this discussion. Actually one of them has been shot down already but I still try to rise it up:

1. Analogy of cases where the expression of measure is between the numeral and the quality, like 1.80 (meter) boy|u|n|da|yım.

2. Ellipsis.

a) There is something that we can’t see between 35 and yaşındayım. A measure. yil, yaş? In this case yaş in yaşındayım wouldn’t mean ‘year’ but ‘age’ (which is its first equivalent in some dictionaries). The structure would be like in the boy thing in nr 1.

b) There is something that we can’t see before 35. A genitive marked noun: something like yaşın, yıllarımın, zamanın… I suggested this but scalpel said in that case the numeral should be the ordinal 35inci. Is it impossible to imagine that it once was ordinal but it was simplified? In other languages I have seen processes like this. It seems like a very logical development because the ordinal marker is nothing but a burden from the point of view of information. But if something like this really happened there should be marks of it in old litteral language, I suppose.

My knowledge is limited but a learner can speculate as much as she wants. If she has the guts.

A simple explanation try of me:

25 yaş-ı-n-da-y-ım

As you know we can use possive -ı if it is a part of noun clouse. There are two options here:

1)Definite noun clause:

If there is a possesive -ı in the sentence it belongs to 3.person. If we put "onun" into this sentence. It doesnt make sense.

2)Indefinite noun clause:

It is also not an idefinite noun clause like kapı kolu, araba tekeri because 25 functions as an adjective here.

Another theory:

If it is 25 (yıl) yaşındayım than it is correct. I looked at google and i found "yıl yaşında"

http://www.google.com.tr/#hl=tr&sclient=psy-ab&q=y%C4%B1l+ya%C5%9F%C4%B1nda&oq=y%C4%B1l+ya%C5%9F%C4%B1nda&aq=f&aqi=&aql=&gs_l=serp.3...595l2779l2l3424l5l5l0l0l0l1l1781l6027l6-2j2j1l5l0.frgbld.&pbx=1&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&fp=d651b3bd1e9eca4c&biw=1138&bih=464

Result: I am not sure anymore.

Super. gokuyum, this is approximately what I suggested in 2a.

It is possible. Because we say:

60 kilo ağırlık-ı-ndayım.

1.70 metre boy-u-ndayım. (We dont say metre often. We say 1.70 boyundayım)

60 santim en-i-ndeyim.

But the question is why we dont say 25 yıl yaşındayım? In English there is yıl (year). I am 25 years old.

So the question is:

Did we use "yıl" before "yaş" in the past? We should examine old texts.

Finally you admit that "xx yaşımdayım" is possible too

A big step..

But I disagree with the translation..

"yaş" means "age" in English..

(Senin) ülkende çocuklar hangi yaşta (or kaç yaşında) okula başlarlar? at what age do children start school in your country?

35 yaşımdayım = I´m in my age of 35 (not I´m in my 35 years)

benim yaşım = my age

senin yaşın= your age

yaşın kaç? (or hangi yaştasın? or  kaç yaşındasın?)= what is your age?

senin yaşında oğlum var = I have a son your age

ben, senin yaşındayken = when I was your age

This is how it is in Turkish:

35 yaşımdayım -- I am in my age of 35

35 yaşındasın -- you are in your age of 35

35 yaşında --s/he is in her/his age of 35

...

yaş = age

(benim) yaşım = my age

(benim) yaşımda = in my age

* benim yaşımda bir adam = a man in my age

* bu yaşımda = in this age of mine

...