Although it is used almost exclusively as a 接尾語（せつびご） (suffix), なし
is actually a noun that means ないこと ‘thing without’. なし
, or 無（な）し when using kanji, will be used after nouns that it is indicating as ‘not having (A)'.
Depending on what the speaker wants to convey, なし
is often partnered with the 助動詞（じょどうし） (auxiliary verb) だ
, or its conjunctive form で. なしで will mean ‘(A) that is without (B)’. When connected directly to another noun, の will be required after なし
. Again, this will mean ‘(A) that is without (B)'.
That way of thinking is without error. (That way of thinking is flawless)
Please don't put up a store in the park without permission.
What is a popular dish without meat?
does not always need to be connected directly to another noun. Often, it can be used in reference to an event or thing that someone wants to skip, or avoid. In cases like this, it may be paired with the 格助詞（かくじょし） (case marking particle) に
, to show determination to avoid something.
Because I ate a lot yesterday, I am going to skip eating lunch today.
Let's avoid inviting our senpai because he kills the mood.
無（な）し is quite different to 抜（ぬ）き, another suffix-like structure that is used to convey ‘without (A)’. The primary difference is in the original meanings of these words. Having a relationship with ない, なし
simply means ‘without’. 抜（ぬ）き comes from the う - Verb 抜（ぬ）く ‘to remove’. Due to this, 抜（ぬ）き implies the intentional removal of something. Often 抜（ぬ）き and 無（な）し are interchangeable, but not always.
Remove the wasabi please.
Without the red peppers please.