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's an example of an abelian group without associativity, inspired by an answer to this question. Consider the game of rock-paper-scissors: $R$ is rock, $P$ is paper, $S$ is scissors, and $1$ is fold/draw/indeterminate. Let $\ast$ be the binary operation "play". 
\begin{array}{r|cccc}
\ast & 1 & R & P & S\\
\hline
   1& 1 & R & P & S \\
   R & R& 1 & P & R\\
   P & P& P & 1 & S\\
   S & S& R & S & 1 .
\end{array}
The mutliplication table above defines a set of elements, with a binary operation, that is commutative and non-associative. Also, each element has an inverse (itself), and the identity exists.
