What's happening in these sentences is that you are starting with an original idea like this:
I have never met a man who is as rich as he is rich.
That sentence sounds strange because we haven't applied any ellipsis- the process of pruning unnecessary or repeated items from a sentence. The minimum ellipsis for a natural sentence is to remove the repeated rich:
I've never met a man who is as rich as he is.
We can also remove the unnecessary who is (this is called whiz-deletion), giving your sentence 3:
- I've never met a man as rich as he is.
If you look at the original sentence, you will see the there are two instances of is, so we can also remove the duplicate is:
I've never met a man who is as rich as he.
Take out the who is and you get to sentence 2:
- I've never met a man as rich as he.
With all that ellipsis, many people will have lost touch with the grammar of
the complete sentence, and feel uncomfortable with the he at the end of the sentence, so they change it to the first one:
- I've never met a man as rich as him.
This is what the majority of people would say in normal speech. When you look at writing (see this NGram), you will find significantly more sentences like 3, and a very small number of examples like sentence 2 - and mainly at the 'literary' end of the market. Meanwhile, sentence 1 hardly figures at all in written English: this gives a pretty good indication that sentence 1 is not generally considered acceptable in writing.
In a formal setting, 3 would be OK in any context and 2 would be acceptable in a literary context (poetry and novels, especially historical novels): 1 would be acceptable if it appeared in dialogue in a novel.
As a side note, the reason that him/he is an issue can best be explained with a simpler and better documented example. The dictionary definition of [than] states that it can be a preposition or a conjunction. As a preposition, **than** requires an object pronoun:
Jane is richer than him
As a conjunction, than requires a clause (which contains a subject pronoun):
Jane is richer than he is
For some reason, we prefer to treat than (and also the as ... as construction) as a preposition in everyday speech, but as a conjunction in formal writing. To quote in full the passage that Mari-Lou referred to in her answer:
Research associated with the Longman Grammar (1999) showed that speakers mostly use than (and as) as prepositions (i.e. with a following object pronoun) and only rarely with a following subject pronoun. Fiction writers make about equal use of the two constructions, while academic writers use neither.