Can this be considered a complete sentence?
There lived a princess named Gretchen.
Can this be considered a complete sentence?
There lived a princess named Gretchen.
No. It's ungrammatical.
The rule of There-Insertion requires some adverbial -- of place, time, or circumstance.
Except, let it be said, as usual, with be, which it most commonly occurs with.
There are some common constructional usages with be that don't need adverbials,
e.g:
- Enumerating lists: There's holmium, and helium, and hafnium, and erbium.
- Asserting existence: There is a number which is the square root of -1.
(often with stressed be).
But other There-Insertion verbs like live require an adverbial. That's the function of story-initial phrases like Long, long, ago or Once upon a time or In a castle above the city -- they establish the time, place, or circumstance of the existence.
One could of course say
But that's similarly incomplete; put an adverbial in
and it's still weird, because English does not prefer indefinite subjects with existential verbs.
That's what There-Insertion is for; it inserts the dummy subject there.
Structurally, sure. It's the same as saying "Princess Gretchen lived".
The {There} {was/is/exists} {something}
structure is usually just a lead-in to a prepositional phrase or subordinate clause, but by definition phrases and subordinate clauses are grammatically optional.
Semantically, you're leaving a big "and then what?" at the end -- but that would normally be a stylistic choice, to draw the reader in.