  How rare are extreme-spin primordial black holes? We show how, from an
underlying distribution of PBH spin, extreme-value statistics can be used to
quantify the rarity of spinning PBHs with Kerr parameter close to 1. Using the
Peaks-Over-Threshold method, we show how the probability that a PBH forms with
spin exceeding a sufficiently high threshold can be calculated using the
Generalised Pareto Distribution. This allows us to estimate the average number
of PBHs amongst which we can find a single PBH which formed with spin exceeding
a high threshold. We found that the primordial spin distribution gives rise to
exceedingly rare near-extremal spin PBHs at formation time: for typical
parameter values, roughly up to one in a hundred million PBHs would be formed
with spin exceeding the Thorne limit. We discuss conditions under which even
more extreme-spin PBHs may be produced, including modifying the skewness and
kurtosis of the spin distribution via a smooth transformation. We deduce from
our calculations that, if indeed asteroid-mass PBHs above the current
observational limit on evaporating PBHs of mass ~10^{17} g contribute
significantly to the dark matter, it is likely that some of them could be
near-extremal PBHs.
