These are the suggestions I found on Google as well, but the point of these is that they’re for polite use in an isolated ‘phone number only’ context. If other questions had already been asked, I think the ‘what’s your number?’ question would be more acceptable.
For what it’s worth, 何番 is definitely the preferred version in my EN-JP dictionary (Wisdom)
▸ 君の部屋の番号は何番ですか
What is the number of your room [your room number]? (!「105号室」なら Room No. 105 で, Room number one-o | oʊ | -five と読む)
and is also what I found in websites on English phrases for asking for phone numbers with translations. (I searched 電話番号を聞く フレーズ.)
However, I think the most telling thing is what I found on this page that’s supposed to be linked to NHK’s language learning resources explaining how you ask for phone numbers in Chinese:
解説
この文の“多少”は「いくつ」「どのくらい」「何番」といった意味で、“~号码是多少?”は、電話番号に限らず“车牌号码是多少?”(車のナンバーはいくつですか)、“房间号码是多少?”(部屋番号は何番ですか)など、様々な番号をたずねる時に応用できます。My translation: The 多少 in this sentence means[, for example,] ‘how many’, ‘about how much/many’, ‘which number in a sequence’, and ~号码是多少?can be used at moments when one is asking for various sorts of numbers e.g. 车牌号码是多少?(‘What’s your car number?’); 房间号码是多少?(‘What’s your room number?’)
In short, yes, 何番 is the natural way. The fact that 何番 is listed as one of the possible meanings of 多少, instead of simply 何, tells us that 何番 is a common, natural phrase in this context.
Just for reference, it appears in Daijirin (which has been taken offline on most sites)
六 動詞の連用形の下に付いて複合動詞をつくる。
① 最後まで…する,…し通すの意を表す。「がんばり―・く」「走り―・く」「耐え―・く」
② ひどく…するの意を表す。「困り―・いている」
and also in Meikyou and Shinmeikai (here’s the one from Shinmeikai):
三〔「動詞連用形+抜く」の形で、接尾語的に〕
㊀積極的な行動を最後までとる。
「やり抜く・がんばり抜く」
㊁すっかり…する。
「困り抜く〔=はてる〕」
So yeah, indeed, in both cases, it’s about doing something to an extreme degree in either a good/neutral way, or a bad way. There’s also a notion of completeness. I’d liken it to ‘up’ in English e.g. ‘round up’, ‘f–k up’, ‘tie up’, which also express such ideas (completeness, completion, extremeness) with a word that suggests detachment (via vertical distance, in this case).




