Non-associative: set with a binary operation, but has inverses and identity

The object you are describing is called a loop. If you want a high powered example that fits in one mouthful, try the multiplicative group loop of non-zero octonions. As a more elementary alternative, I thought I'd give you a finite loop to ponder on (although you may need to work a bit to verify all the properties). I present Loop 8.1.4.0:

$$\begin{array}{c|c|c|c|c|c|c|c|} \circ& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\\hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\\hline 1 & 1 & 7 & 5 & 0 & 6 & 2 & 4 & 3\\\hline 2 & 2 & 6 & 7 & 5 & 0 & 3 & 1 & 4\\\hline 3 & 3 & 0 & 6 & 7 & 5 & 4 & 2 & 1\\\hline 4 & 4 & 5 & 0 & 6 & 7 & 1 & 3 & 2\\\hline 5 & 5 & 2 & 3 & 4 & 1 & 7 & 0 & 6\\\hline 6 & 6 & 4 & 1 & 2 & 3 & 0 & 7 & 5\\\hline 7 & 7 & 3 & 4 & 1 & 2 & 6 & 5 & 0\\\hline \end{array}$$

Here $0$ is the identity, and the inverses are $0,3,4,1,2,6,5,7$ for $0\dots7$ respectively. Consider $1\circ 1\circ 2$ to show non-associativity and $1\circ 2$ for non-commutativity.

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The above is actually an example of a Bol loop, which satisfies the more complicated weak associativity property $a(b(ac))=(a(ba))c$. For general loops, there are smaller examples; the smallest non-associative loop has order $5$ - here is one of them:

$$\begin{array}{c|c|c|c|c|} \circ & 0 & 1 & 2 & 3 & 4\\\hline 0 & 0 & 1 & 2 & 3 & 4\\\hline 1 & 1 & 4 & 0 & 2 & 3\\\hline 2 & 2 & 3 & 4 & 1 & 0\\\hline 3 & 3 & 0 & 1 & 4 & 2\\\hline 4 & 4 & 2 & 3 & 0 & 1\\\hline \end{array}$$

Note that this does not have two-sided inverses, since we have identities like $1\circ 2=0$ and $3\circ 1=0$, so that the left inverse of $1$ is $2$ and the right inverse is $3$. For non-associativity just consider $1\circ 1\circ 1$ (this loop is not even power-associative).