Is there anything like "Saturated Fullerene"?
I'm limiting my answer to $\ce{C60}$-fullerene, but the same idea applies to the higher fullerenes as well. 

In short, it's not possible to fully hydrogenate $\ce{C60}$ to $\ce{C60H60}$. This is a result of the bond strain that would result upon hydrogenation, as the addition of hydrogen across an alkene in $\ce{C60}$ produces carbon atoms that would optimally be tetrahedral with angles near $109.5^\circ$, while the bond angles in $\ce{C60}$ are $108^\circ$ and $120^\circ$ for the pentagonal and hexagonal rings, respectively. 

Partial hydrogenation of $\ce{C60}$ with zinc and hydrochloric acid has been done to form $\ce{C60H36}$,1 but this compound degrades quickly in the presence of oxygen to form hydroxylated groups; degradation of the fullerene cage and formation of ketones also occurs. Efforts to hydrogenate further under high pressure hydrogenation conditions results in complete breakdown of the fullerene cage, with pretty interesting results, including ejection of carbon atoms and $\ce{C2}$ loss, producing hydrogenated species with fewer than 60 carbon atoms.2 Collapse of the fullerene cage was also noted, forming hydrogenated compounds with 25 - 45 carbon atoms. 




Cataldo, Franco. Fullerenes, Nanotubes and Carbon Nanostructures 2003, 11 (4), 295-316. DOI: 10.1081/FST-120025852
Talyzin, Alexandr V., Yury O. Tsybin, Jeremiah M. Purcell, Tanner M. Schaub, Yury M. Shulga, Dag Noréus, Toyoto Sato, Andrzej Dzwilewski, Bertil Sundqvist, and Alan G. Marshall. J. Phys. Chem. A 2006, 110 (27), 8528–8534. DOI: 10.1021/jp0557971