  We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic)
which contains no essential closed surfaces, but for any even integer $g> 0$
there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$,
the 3-manifold obtained by attaching 2-handle to $M$ along $r$, contains an
essential separating closed surface of genus $g$ and is still hyperbolic. The
result contrasts sharply with those known finiteness results for the cases
$g=0,1$. Our 3-manifold $M$ is the complement of a simple small knot in a
handlebody.
