Each atom of the element argon has 18 electrons. How many chemical bonds is one atom of argon likely to make with atoms of other elements?

It won't. Argon is an element found in the Group 8A also called as the noble gases. They are called as such because they do not react with other elements because their electronic configuration is very stable. This is because they already obey the Octet rule, which has to do with eight electrons occupying around the element. So, it is very rare to have reaction involving any of the noble gases.