The hyperessive is the exaggerated counterpart of the inessive case. It denotes an area deep within the root. It can typically be translated into English as deep inside or deep in. It can only be used to refer to physical locations inside of things.
When referring to things which are deep inside of other things, you don’t necessarily always need to use the hyperessive. If the fact that it’s not just inside, but deep inside is not important, then the hyperessive needn’t necessarily be used.
deep inside the cave
deep inside the tree
deep into the sky
Verbal Derivation Implication
When a hyperessive noun is derived into an action verb, it implies a verb of motion that results in the subject being deep inside of the root of the verb. This can be conceptually translated into English as go deep into. The manner of entry/movement is either left unmentioned or clarified with adverbs or additional clauses (usually the former).
When verbed, the hyperessive and the hyperlative cases are identical in purpose/function. Potayto/potahto, y’know? This is true when speaking of both active and stative derived verbs.
I [went] deep into the cave.
Someone [went] deep into China with my cat.
(Lit: Someone‘Chinadeepintoed catmywith.
And as usual, when a hyperessive noun is derived into a stative verb, it is copulaic/existential.
The creature is deep within the cave.