  Physical fractals invariably have upper and lower limits for their fractal
structure. Berry has shown that a particle sharply confined to a box has a wave
function that is fractal both in time and space, with no lower limit. In this
article, two idealizations of this picture are softened and a corresponding
lower bound for fractality obtained. For a box created by repeated measurements
(\`a la the quantum Zeno effect), the lower bound is $\Delta x\sim \Delta t
(\hbar/{mL})$ with $\Dt$ the interval between measurements and $L$ is the size
of the box. For a relativistic particle, the lower bound is the Compton
wavelength, $\hbar/mc$. The key step in deriving both results is to write the
propagator as a sum over classical paths.
