One per reaction, unless a feature says otherwise
For instance, the fighting style Tunnel Fighter (from UA: Light, Dark, Underdark!) says:
you can make opportunity attacks without using your reaction
Since you don't use your reaction in this case, you can make several OAs per turn with Tunnel Fighter. Otherwise, you make only one or zero OAs per turn, as NautArch said, providing you still have your reaction and you can use it (there are several abilities, preventing creatures from taking reactions, like the Shocking Grasp spell).
There are also "OA-like" attacks, like the Sentinel feat's third benefit (PHB, p. 169-170):
When a creature within 5 feet of you makes an attack against a target other than you (and that target doesn't have this feat), you can use your reaction to make a melee weapon attack against the attacking creature
But it also expends your reaction (hence, can be done only once), despite not being an opportunity attack, and it does not trigger any "when you hit a creature with an opportunity attack" features.