Curry fixed-point combinator ($Y$)

With $A_f=\lambda x.f(xx)$, the combinator $Y=\lambda f.A_f{A}_f$ satisfies $Yf{\equiv }_{\beta }f(Yf)$. Every finite reduct of $Y$ is, up to alpha equivalence, $\lambda f.f^n(A_f{A}_f)$: the remaining copy of $A_f{A}_f$ is its only beta-redex.