I came across this: いつのいつまで見ぬ振りしようか According translation sites, the double いつ means "until when," but I would like to know the reason behind the repetition.
I believe this is what Japanese dictionaries would describe as 「いつまで」を強めた表現, that is, 'how long' is being emphasized, something like English, 'for heaven's sake how long do I have to pretend not to notice?', said out of exasperation or frustration, for example.
You will also find いつ時, いつどき, used in a similar sense, a very emphatic form of いつ.
の as usual, is the copula, hence 'until what time [which is] when'. No, this is not a normal utterance, but you can see how the meaning is derived, if you postulate some underlying structure like this, so syntactically, the sentence is well-formed.
If this really needs to be analyzed, this answer may help. Two similar words (including na-adjectives) can be joined with の for emphasis. There is also the set phrase 今の今まで. So this いつのいつ can be understood as an emphatic version of いつ.
However, いつのいつ is not standard Japanese, and I would say it does not contain any useful patterns that would be valuable to learn, even as slang or something. The reason the word is repeated like this is simply "because they're lyrics". The singer might have thought saying いつ twice would improve the rhythm, but there is nothing more profound than this.