I have no choice but verb(to / -ing or none)

Question

Which one is correct?

1. I have no choice but to kill you.

2. I have no choice but killing you.

3. I have no choice but kill you.

or do they have different meanings?

Answer 1

But in these examples functions as a preposition meaning except. It is sometimes followed by a verb.The form depends on what came before. Infinitives are usually without to, but sometimes we need a gerund.

She does nothing but sleep all day.

He doesn't like anything but playing on his computer. (like ... playing)

Infinitives with to are used after no choice, option, alternative but

I have no choice but to kill.(This sentence is correct).

Answer 2

So your question is -

...no choice but [?]

What can take the place of [?]. That is what you are asking, right?

Well, but there is a preposition, meaning except, and it can license either a matrix-licensed-complement or a Noun Phrase [NP].

1.(a) There’s been nothing but trouble since he came. [License NP]
1.(b) Black people used to have no choice but field work because there were few other opportunities to earn money. [License NP]

2.(a) We had no choice but to believe them. [Matrix clause - We had (to believe them)]
2.(b) He had no choice but to ​leave. [Matrix clause - He had (to leave)]

So if it can license a NP, there is no reason for it to license Gerund-Participle. So grammatically the following has to be correct -

I have no choice but killing you.

But I must make it a point here that this construction is very rare.

So let's summarise it -

3.(a) I have no choice but to kill you. (CORRECT)
3.(b) I have no choice but killing you. (CORRECT but very rare)
3.(c) I have no choice but kill you. (Doubtful. I would go as far as saying that it's incorrect, though I have some example sentences with similar constructions in CORPUS even more than the structure depicted in #3(b)]