"What next?" Why don't people use "is" here?

“What next?” would be an informal way of speaking in my opinion. You could say “what is next?” but it sounds robotic. I would use “what’s next?” with a contraction of “what” and “is.”

"What next?", "What's next?", and "What is next?" are all fine. You don't really need the "is", but you might include it in more formal writing.

What now?

I think it is informal and would not be appropriate in a formal setting. Like "How you doing" instead of "How are you doing"

both forms are correct, but “what next” is informal and “what’s next” sounds natural while “what is next” often sounds robotic or stilted

I think in my mind when I say “What next?” the “what” is a placeholder for whatever the answer is. Like a blank space I’m asking you to fill in. If that makes sense. Like if we’re baking a cake and I think it’s time to add the eggs I might say “Eggs next?” But if i don’t know what should come next I’ll just openly ask what should go in that “blank space” so to speak. I’m not sure how you would describe this grammatically, but I’m a US native speaker.

Next is being used as an adverb here. Consider "What are we doing next" or "What's up next". In both cases "is" is not modifying "next" directly. Now of course, "What's next" sounds perfectly natural, but "next" is playing a different grammatical rule than in "What next". In "What's next", 'next' is an adjective, and in "What next" it's an adverb

"What is next" isn't technically wrong, but in day to day conversation it's almost always replaced by the contraction "What's next" or just outright "What next". There are slight differences in formality for them, but not to the point that anyone would really care (for example, if it was a very informal situation, or someone was annoyed/exasperated, they'd probably opt to go for "what next". I would be probably a little thrown off if I heard "What is next" (it seems a little bit robotic).

It's colloquial as others have said but usually it introduces a "slippery slope". "He gets up, then we all get up; what next? Anarchy!?!" (I know that's not the actual line in the breakfast club but it was useful to adapt for an example).

Such snippets of sentences without verbs are very common, eg what now, what then, why now, why today, why not, how then, who with, etc.

It’s a fixed phrase.

"What next" is not informal, as many of the comments here misleadingly claim. Nor is it simply "what's next" but with the 's omitted.

It’s a short form expression (not strictly correct, but commonly used). As others said, *is* is usually the missing word between *what and *next*, but could also be “what happens next”, “what comes next”, “what do we do/should we do next”, etc.

One of English's many exceptions.

I usually hear/say, “What’s next.” See also, this great book about a great TV show https://a.co/d/imH3XXl

Sentence fragments and 'telegraphic' communication (where unnecessary words are omitted) are idiomatic in speech across many registers. Some are perfectly acceptable in all but the most formal of writing. Removing the verb allows for both directness and ambiguity. It's a quick way to float an open question which could mean (amongst other things): What is next? What might be next? What should be next? What was going to be next?