Large bosonic or continuous variable nonlinearities can have numerous applications, ranging from the generation of cat states for quantum computation, through to quantum sensing where the sensitivity exceeds Heisenberg scaling in the resources. However, the generation of ultra-large nonlinearities has proved immensely challenging experimentally. We describe a novel protocol where one can effectively generate large Kerr-type nonlinearities via the conditional application of a linear operation on an optical mode by an ancilla mode, followed by a measurement of the ancilla and corrective operation on the probe mode. Our protocol can generate high-quality Schrödinger cat states useful for quantum computing and can be used to perform sensing of an unknown rotation or displacement in phase space, with super-Heisenberg scaling in the resources. We finally describe a potential experimental implementation using atomic ensembles interacting with optical modes via the Faraday effect.