Band-limited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals: they can ‘superoscillate’. In physics, this counterintuitive mathematical phenomenon is associated with almost-destructive interference, and occurs near phase singularities in optics and on the world’s ocean tides; and it is associated with quantum weak measurements. Where superoscillations occur, functions are exponentially weak and vulnerable to noise. They are an unexpectedly compact way of representing fractals. Superoscillations in red light can escape as gamma radiation.