  Bosonic fields on rotating black hole spacetimes are subject to amplification
by superradiance, which induces exponentially-growing instabilities (the `black
hole bomb') in two scenarios: if the black hole is enclosed by a mirror, or if
the bosonic field has rest mass. Here we present a time-domain study of the
scalar field on Kerr spacetime which probes ultra-long timescales up to $t
\lesssim 5 \times 10^6 M$, to reveal the growth of the instability. We describe
an highly-efficient method for evolving the field, based on a spectral
decomposition into a coupled set of 1+1D equations, and an absorbing boundary
condition inspired by the `perfectly-matched layers' paradigm. First, we
examine the mirror case to study how the instability timescale and mode
structure depend on mirror radius. Next, we examine the massive-field, whose
rich spectrum (revealed through Fourier analysis) generates `beating' effects
which disguise the instability. We show that the instability is clearly
revealed by tracking the stress-energy of the field in the exterior spacetime.
We calculate the growth rate for a range of mass couplings, by applying a
frequency-filer to isolate individual modal contributions to the time-domain
signal. Our results are in accord with previous frequency-domain studies which
put the maximum growth rate at $\tau^{-1} \approx 1.72 \times 10^{-7}
(GM/c^3)^{-1}$ for the massive scalar field on Kerr spacetime.
