Let $M$ be an irreducible oriented compact 3-manfold. Take a maximal collection of disjoint non-parallel essential disks and essential annuli (in fact this collection is finite). Then cut $M$ along these surfaces, and call the new manifold by $M'$. Does there still exist essential disks or essential annuli?
It seems the answer is no because what is cut is a maximal collection of such surfaces, but even if $M'$ contains an essential annulus(disk), it doesn't mean this annulus(disk) is obtained by cutting an essential annulus(disk) in $M$. For example, an annulus can be obtained by cutting a disk with 2 holes along an arc.