I seem to recall once hearing a result to the effect that $\emptyset^{(\omega)}$ was the double jump of some other degree, but could not be the triple jump of any degree. However I'm unable to find the exact result. Does anyone know what I might be thinking of (or what is actually known about jump inversion on $\emptyset^{(\omega)}$, if I'm remembering this completely wrong)?
This doesn't exactly answer your question but... If $A$ is any upper bound for the arithmetic degrees then $0^{(\omega)}$ is recursive in $A^{\prime\prime}$. Enderton and Putnam proved that there upper bounds with $A^{\prime\prime}=0^{(\omega)} $ Dave 


As far as actual jump inversion goes, any degree $X \geq 0^{(n)}$ is the $n$th jump of some other degree. An easy way to see this is to apply Friedberg jump inversion relative to $0^{(n1)}$, then relative to $0^{(n2)}$, and so on down to $0$. The theorem is also true through transfinite iterates of the jump: if $X \geq 0^{(\alpha)}$, then $X$ is the $\alpha$th jump of some degree. The general version of this theorem for any $\alpha$REA operator is due to Jockusch and Shore (1984). 

