  A key physical quantity during reionization is the size of HII regions.
Previous studies found a characteristic bubble size which increases rapidly
during reionization, with apparent agreement between simulations and analytic
excursion set theory. Using four different methods, we critically examine this
claim. In particular, we introduce the use of the watershed algorithm -- widely
used for void finding in galaxy surveys -- which we show to be an unbiased
method with the lowest dispersion and best performance on Monte-Carlo
realizations of a known bubble size PDF. We find that a friends-of-friends
algorithm declares most of the ionized volume to be occupied by a network of
volume-filling regions connected by narrow tunnels. For methods tuned to detect
the volume-filling regions, previous apparent agreement between simulations and
theory is spurious, and due to a failure to correctly account for the window
function of measurement schemes. The discrepancy is already obvious from visual
inspection. Instead, HII regions in simulations are significantly larger (by
factors of 10-1000 in volume) than analytic predictions. The size PDF is
narrower, and evolves more slowly with time, than predicted. It becomes more
sharply peaked as reionization progresses. These effects are likely caused by
bubble mergers, which are inadequately modeled by analytic theory. Our results
have important consequences for high-redshift 21cm observations, the mean free
path of ionizing photons, and the visibility of Ly-alpha emitters, and point to
a fundamental failure in our understanding of the characteristic scales of the
reionization process.
