  For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow
N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds
$(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded
$L^2$ curvature energy into a compact Riemannian manifold $(N,h)$ with
uniformly bounded biharmonic energy, we establish a compactness theory modular
finitely many bubbles, which are finite energy biharmonic maps from
$\mathbb{R}^4$, or from $\mathbb{R}^4 / \Gamma$ for some nontrivial finite
group $\Gamma \subset SO(4)$, or from some complete, noncompact, Ricci flat,
non-flat ALE 4-manifold (orbifold). To achieve this, we develop a sophisticated
asymptotic analysis for solutions over degenerating neck regions.
