Hi,
I am looking for at least three examples of a set S and and operation * such that there is a left identity and no right identity.
Just make some up!
For instance, let $\displaystyle S={a, b}$ and * be such that
$\displaystyle a*b = b$
$\displaystyle b*a = a$
$\displaystyle a^2 = a$
$\displaystyle b^2 = b$.
This is the right-zero semigroup of order 2. Note that every element is a left identity.
You should also notice that if $\displaystyle S$ has a right and a left identity then they must be the same element, a `common or garden' identity. So, each of your examples must have a left identity which is not a right identity, which is much easier to verify!
(This post would have been better suited to the algebra forum...)