  We develop a theory of "minimal $\theta$-graphs" and characterize the
behavior of limit laminations of such surfaces, including an understanding of
their limit leaves and their curvature blow-up sets. We use this to prove that
it is possible to realize families of catenoids in euclidean space as limit
leaves of sequences of embedded minimal disks, even when there is no curvature
blow-up. Our methods work in a more general Riemannian setting, including
hyperbolic space. This allows us to establish the existence of a complete,
simply connected, minimal surface in hyperbolic space that is not properly
embedded.
