The authors study the limit behaviour as \(t\to \infty\) of the process \[ \xi(t)=\sup \{s :\;e\leq s\leq t,\quad W(s)\geq (2s \log \log s)^{1/2}\}, \] where \(W(t)\) is a Wiener process. The main result is the following Theorem: \[ \liminf_{t\to \infty}[\frac{\log \log t)^{1/2}}{(\log \log \log t)\cdot \log t}]\log \frac{\xi (t)}{t}=-C\quad a.s., \] where \(C\) is a positive constant and \(2^{-2}\leq C\leq 2^{14}\).