What properties of an element determines the maximum number of bonds it can make?
Considering the number of other atoms a central atom can be bonded to, "hybridization [is] irrelevant" and "the coordination number ... is not restricted by the number of
hybrid orbitals that the atom can form", quoting On the role of d orbitals in sulfur hexafluoride J. Am. Chem. Soc., 1986, 108 (13), pp 3586–3593.  

Examples such as $\ce{CLi6}$ and $\ce{TeF8^{2-}}$ are cited and emphasis is placed upon the ionic character of the bonding by the cited reference.

In my opinion, the main limiting factor is how many atoms can simultaneously be close to the central atom, this increasing with increasing radius of the central atom and with decreasing radius of the surrounding atoms.