I think 8, 4 to the sides and 4 diagnle
How many chess pieces can be defending a square at once?
What do you define as "normal"? No promoted pieces? Anyways, here's an attempt without the normal requirement...

I count 16. The diagram below shows 16 pieces covering e3 directly. Of course, we could add a number of rooks, but they wouldn't directly cover e3.

I think direct covering is straightforward: vector attacks ( same rank, file or diagonal ) on a square can come from eight directions, knight attacks can also come from eight squares and these squares do not block vector attacks, because knights are designed to be able to move to the closest squares not on the same rank, file or diagonal. This is 16, no question about that.
The non trivial question is how many attacks are possible if both direct and indirect attacks are allowed.

Here is my take on the case where both direct and indirect attacks are allowed.
How many pieces can attack a square ( piece is non king, non pawn )?
It is known that a legal chess position can have at most 26 pieces.
However the square under attack is either a dark square or a light square making one of the original bishops of both sides useless.
If both sides keep their original bishops, this would make for 24 pieces that can attack.
At the cost of having one less piece than 26, both sides can turn their dark square bishops into lights square bishops or vica versa.
This means that at most 25 non king, non pawn pieces can attack a square in a legal position.

What SmyslovFan said.
Edit: But if the square isn't _attacked_, can we really say it is _defended_? hmmmmmmmm??

I think piece has a definite meaning in chess. It is non king, non pawn.
The term which includes kings and pawns is 'chessman'.
So if the question intended to include king and pawn attacks, it should have asked 'How many chessmen can be defending a square at once?'

Direct and indirect attacks are both attacking a square.
If two rooks are lined up on a square containing a pawn, you would say you had two pieces defending the pawn, not one direct and one indirect.
My answer was correct, as if both sides have pieces hitting a square, one is attacking it, one is defending ;)
Let's see who gets it right ;)
EDIT: Promoted pieces can be used! I apologize!