Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F:
x*y = x+y - xy/k
Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element.
Moreover if you have an element a, such that 0 < a < k, then
a < a * a < a * a * a < ....< a^n < a^{n+1} < ...... < k.
This is why I think of it as "Zeno's Monoid". However when I search for examples of monoids, I never find it. Does anybody know of a reference?