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For each real $k>0$, \begin{equation}E\psi_\infty(|X|/k)=\infty\,P(|X|>k)+P(|X|=k) \\ =\left\{\begin{aligned}\infty\text{ if } P(|X|>k)>0,\\ P(|X|=k)\le1\text{ if } P(|X|>k)=0. \end{aligned}\right. \end{equation} So, indeed, $\|X\|_{\psi_\infty}<\infty$ iff $X$ is essentially bounded. Moreover, $\|X\|_{\psi_\infty}=\text{ess}\,\text{sup}\,|X|$.