When I listen to the word: "御幾つ", of course I can quickly wrote down it in pure hiragana: おいくつ. The main problem is that how can I choose to write the kanji and which one (or all) is/are OK?
For me, "御" is easy to write but "幾" is too difficult.
I wonder whether there are situations that I must write all the kanji (especially in exams where I can only write by hand) in Japan?
Or can I choose to write kanji that I am familiar with?
The Japanese government, more specifically the 官公庁{かんこうちょう}, has a set of regulations regarding official government texts.
If we take a look at 1-(2)-ウ, we can see that the official rule is that if the word following 御/お/ご is in 漢字, then we should write 御, however if the word following it is in 平仮名, we should write お or ご, depending on whether the word is of Japanese or Chinese origin.
However personally, I and many of the people surrounding me don't necessarily follow the first part of this rule regarding 御+漢字; we aren't the Japanese government after all, we don't have to.
So you'll very often see 平仮名の「ご」 followed by a 熟語, for example ご案内. This gives a softer impression than 御案内, which would much more likely be seen in an extremely formal situation.
This is due in part to the fact that too many long runs of 漢字 that aren't proper names can make a given text look bogged down and hard to read, even if that isn't actually the case.
The art of balancing the amount of 漢字 with 平仮名 and white space to be more pleasing to look at is actually something taught in a lot of grant-application workshops! (I don't know about other fields)
Also if a word (e.g. 幾つ, the polite お〜になる, or the polite ご/お〜ください) has any 送り仮名 or is commonly written in only 平仮名, it is much much more common to write お over 御, even when that word has 漢字 in it as well.
So to answer your specific question, while both おいくつ and お幾つ would be just fine, 御幾つ looks extremely formal/almost archaic (I can't recall ever having seen it written this way though), and 御いくつ just looks wrong.
