  The statistical behavior of the size (or mass) of the largest cluster in
subcritical percolation on a finite lattice of size $N$ is investigated (below
the upper critical dimension, presumably $d_c=6$). It is argued that as $N \to
\infty$ the cumulative distribution function converges to the Fisher-Tippett
(or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably
normalized). The mean grows like $s_\xi^* \log N$, where $s_\xi^*(p)$ is a
``crossover size''. The standard deviation is bounded near $s_\xi^*
\pi/\sqrt{6}$ with persistent fluctuations due to discreteness. These
predictions are verified by Monte Carlo simulations on $d=2$ square lattices of
up to 30 million sites, which also reveal finite-size scaling. The results are
explained in terms of a flow in the space of probability distributions as $N
\to \infty$. The subcritical segment of the physical manifold ($0 < p < p_c$)
approaches a line of limit cycles where the flow is approximately described by
a ``renormalization group'' from the classical theory of extreme order
statistics.
