In this question, I will be working in ZF.

Let $h(\kappa)$ for a cardinal $\kappa$ (not necessarily an ordinal) be the smallest ordinal $\alpha$ such that there is no surjection from a set of size $\kappa$ onto $\alpha$.

$h(\mathfrak{c})$ is defined as $\Theta$.

Let an uncountable ordinal $\alpha$ be a high limit ordinal if and only if for any cardinal $\kappa$ with no surjection onto $\alpha$, there is no surjection of $2^\kappa$ onto $\alpha$.

Equivalently, for any cardinal $\kappa$ such that $h(\kappa)\leq\alpha$, $h(2^\kappa)\leq\alpha$.

Under AC, the high limit ordinals are precisely those which are strong limit cardinals.

Questions: Under AD, what properties does the smallest high limit ordinal have? Can $\Theta$ be the smallest high limit ordinal in ZF? Is it possible in ZF that there exists no high limit ordinal?