[Bublik]

Why is C-C bond energy stronger than F-F bond energy? 

C-C bond energy is 154kJ/mol, F-F bond energy is 142kJ/mol according to a source.

I thought F-F bond energy would be larger, because the radii or the F atoms are smaller, and more of the nucleus is available to grab hold of the other F atom's electrons.

[Corribus]

Bond energy is a function of (1) orbital overlap, (2) Coulombic forces - attraction and repulsion, and (3) electron exchange, which is a purely quantum mechanical effect. Fluorine has a lot more electron density surrounding it than carbon, and therefore a lot more charge repulsion. For the same reason, O-O and N-N single bonds are pretty unstable.