We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in
[0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal,
$f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll}
1-K_-|x-c|+o(|x-c|)& \mbox{ for }x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta)
&\mbox{ for }x>c, \end{array}\right. $$ where we assume that $\beta>1$. We show
that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and
develop a renormalization theory for these maps. The scalings of the
renormalization intervals of the $2^\infty$ map turn out to be
super-exponential and non-universal (i.e. to depend on the map) and the
scaling-law is different for odd and even steps of the renormalization. The
conjugacy between the attracting Cantor sets of two such maps is smooth if and
only if some invariant is satisfied. We also show that the
Feigenbaum-Coullet-Tresser map does not have wandering intervals, but
surprisingly we were only able to prove this using our rather detailed scaling
results.