Does standard English include non-trivial sentences which admit no verb?

One of the interesting aspects of the Maori language, as I understand it, is that it includes a class of sentences which not only have no verb, it is positively wrong to form them with a verb.

Does standard English include any non-trivial sentences which have no verb, and are not simply ellipses of sentences which do include a verb?

African-American Vernacular English includes sentences like, "Where you at?", that have no verb. The corresponding sentence, "Where are you at?", doesn't sound like African American Vernacular English. Adding the verb "are" seems to transform it into White American English.

Phrases like, "Hey!" are interjections, and don't meet my threshold of being non-trivial. Answers like "Tomorrow." to "When will you file your story?" seem to be ellipses; in this case, of the verb-laden "I will file my story tomorrow."

I found a related question here in in EL&U, Is a sentence always grammatically incorrect if it has no verb?. The title notwithstanding, the question actually seemed to be asking for an explanation why a certain phrase was in fact not grammatical.

I realise that "standard English" is a bit of a contradiction. Sorry. I refer to any of the mainstream English language dialects, steering clear of pidgins and creoles, where no doubt lurk all manner of interesting grammatical structures.

One can always use a sentence fragment to stand for a sentence, or part of a sentence, in context. For instance, the utterance (NB - all sentences are utterances, but not vice versa)

• Roasted, just like me.

can hardly be considered a sentence, but as an answer to the question

• How does she prefer her chicken?

it's completely unobjectionable. It's vital to distinguish speech from writing here.

Most people, most of the time, speak English as bursts of familiar phrases (many worn down by deletion of markers and other identifying grammatical chunks), and not as full sentences. Native speakers can often reconstruct speech as full sentences, but not all, and not always, which is why writing is such hard work for many people.

And since speech overwhelms reading or writing by orders of magnitude, speech is where the real language is, and what the grammar refers to.

In ‘The Cambridge Encyclopedia of the English Language’, David Crystal distinguishes between major and minor sentences. A major sentence is:

A type of sentence which is highly productive, such as those with a subject plus predicate structure; contrasts with minor sentence, where the structure lacks some of the constituents found in the major type.

Alternatively, the term non-clausal material is available to describe ‘the parts of a text or discourse which do not consist of clauses’, where a clause is ‘a key structural unit of grammar, normally consisting of a verb phrase plus other elements: subject, object, predicative, adverbial’ (‘Longman Student Grammar of Spoken and Written English’).

The answer to your headline question is that Standard English can, and frequently does, include structures of this kind, particularly in spoken language. Their use in written language depends very much on the context, and in particular on the degree of formality required.