Can one start a sentence with the word apparently?
Apparently he did not pay him back.
I know that one should not start a sentence with because, but what are some words that one shouldn't start sentences with?
There is nothing wrong with starting a sentence with apparently. Apparently is completely different than because in this respect. Because is a subordinating conjunction, which connects a dependent clause to its independent clause. Apparently is simply an adverb, or arguably even an interjection, but it does not under any circumstances connect two clauses.
There is nothing grammatically wrong with starting a sentence with because, as the dependent clause can come before the independent one, e.g:
Because his mother was allergic to dogs, the boy was without a pet for most of his childhood.
This is perfectly valid. There are no words that are universally bad to start a sentence with. In situations where it is wrong to start a sentence with a certain word, it is simply as a side effect to some other rule, or the rest of the sentence can be re-worked in some way for it to be correct.
There's no wrong in starting a sentence with "apparently". Just for knowledge purposes:
Apparently is an adverb. Apparent is an adjective.
Like what the previous comments were: matter of style and preference.