The expression
なん + counter +
か in Japanese is a way of highlighting an uncertain number of something, and comes across similarly to ‘some’, ‘several’, or ‘a few’ in English. Literally, the meaning is much closer to ‘what + counter + I wonder?’, as it shows the speaker's uncertainty about the number.
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そこには何（なん）回（かい）か行（い）ったことがあると思（おも）う。
I think I have been there several times.
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ごめん、紙（かみ）を何（なん）枚（まい）かちょうだい。
Sorry, may I have some paper?
In addition to なん; いく, from the kanji 幾（いく） (how many/much) may be used as part of this construction. While both are fairly common, 幾（いく） is more common as part of set expressions, like
幾（いく）ら (how much) and
幾（いく）つ (how many/old), while なん is more common for counting specific things, like people.
Both 何（なん） and 幾（いく） are regularly seen in plain hiragana and their kanji forms, so it is worth remembering both.
Despite having the same kanji as 何（なに）, なん will almost always be used to count things, while なに will be used to identify things.
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いらない本（ほん）を何冊（なんさつ）か妹（いもうと）にあげた。
I gave several books that I didn't need to my sister.
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お前（まえ）は何者（なにもの）だ！
Who are you! (What kind of person are you)