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, $\|X\|_{\psi_\infty}=\text{ess}\,\text{sup}\,|X|$.