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Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ${\psi }_p\left(x\right)={exp\left(\right|x|}^p)-1$ with 0 < p ≤ 2) of $ma{x}_{0\le t\le \tau }\left|B_t\right|$ or $|B_{\tau }|$ to be finite, where $B={\left(B_t\right)}_{t\ge 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}<\infty$ as soon as $E\left({\tau }^k\right)=O\left(C^k{k}^k\right)$ for some constant C > 0 as k → ∞ (or equivalently $\parallel \tau {\parallel }_{{\psi }_1}<\infty$). In particular, if τ ∼ Exp(λ) or $\left|N\right(0,{\sigma }^2\left)\right|$ then the last condition is satisfied, and we obtain $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}\le K\surd E\left(\tau \right)$ with some universal constant K > 0....