Non-associative, non-commutative binary operation with a identity Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element? 

Here are a couple easy examples, I'll leave it to you to verify their properties.
The natural numbers where $n\ast m=n^m$.
The integers where $n\ast m=n-m$.
The real numbers without zero where $x \ast y = \frac{x}{y}$.
