How well do we need to measure the Higgs boson mass and self-coupling?

Much of the discussion regarding future measurements of the Higgs boson mass and self-coupling is focussed on how well various collider options can do. In this article we ask a physics-based question of how well do we need colliders to measure these quantities to have an impact on discovery of new physics or an impact in how we understand the role of the Higgs boson in nature. We address the question within the framework of the Standard Model and various beyond the Standard Model scenarios, including supersymmetry and theories of composite Higgs bosons. We conclude that the LHC's stated ability to measure the Higgs boson to better than 150 MeV will be as good as we will ever need to know the Higgs boson mass in the foreseeable future. On the other hand, we estimate that the self-coupling will likely need to be measured to better than 20 percent to see a deviation from the Standard Model expectation. This is a challenging target for future collider and upgrade scenarios.