  Infinitely renormalizable H\'enon-like map in arbitrary finite dimension is
considered. The set, $\mathcal N$ of infinitely renormalizable H\'enon-like
maps satisfying the certain condition is invariant under renormalization
operator. The Cantor attractor of infinitely renormalizable H\'enon-like map,
$F$ in $\mathcal N$ has {\em unbounded geometry} almost everywhere in the
parameter space of the universal number which corresponds to the average
Jacobian of two dimensional map. This is an extension of the same result in
$\mathcal N$ for three dimensional infinitely renormalizable H\'enon-like maps.
