  We use Euclidean path integrals to explore the set of bulk asymptotically AdS
spacetimes with good CFT duals. We consider simple bottom-up models of bulk
physics defined by Einstein-Hilbert gravity coupled to thin domain walls and
restrict to solutions with spherical symmetry. The cosmological constant is
allowed to change across the domain wall, modeling more complicated
Einstein-scalar systems where the scalar potential has multiple minima. In
particular, the cosmological constant can become positive in the interior.
However, in the above context, we show that inflating bubbles are never
produced by smooth Euclidean saddles to asymptotically AdS path integrals. The
obstacle is a direct parallel to the well-known obstruction to creating
inflating universes by tunneling from flat space. In contrast, we do find good
saddles that create so-called "bag-of-gold" geometries which, in addition to
their single asymptotic region, also have an additional large semi-classical
region located behind both past and future event horizons. Furthermore, without
fine-tuning model parameters, using multiple domain walls we find Euclidean
geometries that create arbitrarily large bags-of-gold inside a black hole of
fixed horizon size, and thus at fixed Bekenstein-Hawking entropy. Indeed, with
our symmetries and in our class of models, such solutions provide the unique
semi-classical saddle for appropriately designed (microcanonical) path
integrals. This strengthens a classic tension between such spacetimes and the
CFT density of states, similar to that in the black hole information problem.
