Is it possible to start a grammatically-correct English sentence with the word "Than"?

Bob's fat is so much more adorable than everybody else's, Mary said.

-Than everybody else's? You can't be serious.

'I'm very serious. Even more adorable than a- a- a-', halted Mary.

-'Than a what?'

'Than a blue whale on a trapeze.'

Only if the sentence is a partial one, and you are using ellipsis, as in dialogue. In formal writing, it's not proper.

Justin Greer has already given an excellent answer, but it’s worth looking at why some examples of this seem more marked/forced, while others (like W2’s comment on the question) seem rather more plausible.

The most obvious way to get than at the start of a full declarative sentence is to use a “PP-fronting” construction, i.e. putting the prepositional phrase “than …” at the start, where you would normally expect to find the subject of the sentence.

So the key is to notice when and why English uses PP-fronting. It gets used mainly for topicalisation: that is, taking a phrase which would not normally the main topic of the sentence, and making it the topic. (See: the topic–comment model; and a couple of papers on PP-fronting.)

A sentence that fronts a “than…” phrase, then, is going to sound more natural if there is a clear reason for the phrase to get topicalised. One very strong natural reason is if it’s being contrasted with a parallel phrase in another sentence, where the rest of the sentence stays the same.

Beethoven is perhaps a greater composer than Mozart. Than Bach, though, he is certainly not greater!

Another way to get than to the front is to have the subject of the sentence a noun phrase which, by ellipsis, begins with than:

Running faster than a cat is easy. Than a dog, though, is more difficult.

Here the subject is the noun phrase “[running faster] than a dog”. So this example does rely crucially on its context, with the previous sentence supplying the ellipsis. It’s still a fully grammatical simple declarative sentence, though!

"Than" is a word that is normally difficult to start a grammatically correct sentence with.

Also:

Than a bear, the cub is smaller.

Than a more typical sentence structure is this example certainly stranger. However, it is not invalid.

With other conjunctions and prepositions may we make the same construction.

From this point forward shall my answer be deemed complete.

The answer to the question hinges in the definition of the word "phrase". A phrase can be any conceptual expression of some kind of clause, whether grammatically correct or not:

A small group of words standing together as a conceptual unit, typically forming a component of a clause:

‘to improve standards’ is the key phrase here

is a phrase.

So, a more complete answer than this would tell you that "Than" can be used in phrases of a certain grammatical type ONLY, and not phrases of another grammatical type. So perhaps you should ask what kind of phrase cannot be started with the word "Than". That is a truly technical and interesting question for an advanced English teacher.

You may find that it can't be used in to start a "sentence" in it's strict sense because a sentence is:

"A set of words that is complete in itself, typically containing a subject and predicate, conveying a statement, question, exclamation, or command, and consisting of a main clause and sometimes one or more subordinate clauses."