What is the maximum number of pi bonds that can exist in a given pair of atoms?

The maximum number of pi bonds that can exist in a given pair of atoms is 4.
The maximum number of that can exist in a given pair of atoms is 4. When nobles gases are making a bond, they can share all of their eight electrons.
But it depends on your definition. If you think of C-C bonds, you say that there are only two pi bonds, as in acetylene, H-C≡C-H.
This agrees with the molecular orbital picture, which predicts two pi bonds.
You might predict that F₂ has only a single sigma bond. Yet, according to molecular orbital theory, it also has four pi bonds. Two of these are bonding, and two are antibonding.
For Cl₂, molecular orbital theory predicts eight pi bonds — four from the 2p orbitals and four from the 3p orbitals.
They do not contribute to the bond order of the molecule, because they have equal numbers of bonding and antibonding electrons. But they are still pi bonds.
So it all depends on your model and your definition.