  In this paper we consider the winding number, $\theta(s)$, of planar Brownian
motion and study asymptotic behavior of the process of the maximum time, the
time when $\theta(s)$ attains the maximum in the interval $0\le s \le t$. We
find the limit law of its logarithm with a suitable normalization factor and
the upper growth rate of the maximum time process itself. We also show that the
process of the last zero time of $\theta(s)$ in $[0,t]$ has the same law as the
maximum time process.
