Is it possible to write a sentence without a noun? If so, what is an example?
Yes, such as "Go over there."
Yes, we can create sentences without nouns.
One way to do this is to create a sentence that describes an action, or gives an action, without stating who is doing the action. So we can say:
"Come here now!"
These are sentences composed of verbs and adverbs - there are no nouns in them.
These are imperative sentences. An imperative sentence may not even have a subject. The subject of an imperative sentence can be implied. In these examples, the implied subject is the pronoun "you".
And as answered previously, if you don't count pronouns as nouns, then there are even more ways to create sentences without nouns.