= Solution
Take the <Beth number>
$$
\boxed{\kappa=\beth_\omega=\sup_{n<\omega}\beth_n,\qquad\beth_0=\aleph_0,\quad\beth_{n+1}=2^{\beth_n}.}
$$
<Cantor's theorem> makes this sequence strictly increasing. It is a countable cofinal sequence in $\kappa$, while no finite set is cofinal in the <limit ordinal> $\kappa$. Hence $\operatorname{cf}(\kappa)=\aleph_0$. Consequently
$$
\boxed{2^{\operatorname{cf}(\kappa)}=2^{\aleph_0}=\beth_1<\beth_\omega=\kappa.}
$$
Thus <Beth omega is a singular strong limit above the continuum>. No <Continuum hypothesis> is needed for the example.
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