There is no limit to how many times a Pokémon can call for any ally. For example, one user on Reddit chained 100 Rockruff calls for an ally to get a shiny Rockruff.
The reason why calls for ally are unlimited is because it is the new method of chaining Pokémon to obtain Pokémon with increased chances of being shiny, having better IVs, or potentially having their Hidden Abilities.
From the same Serebii link as above, there are a few strategies to improve your chance of having a Pokémon call for help:
The first is to simply keep the Pokémon at a low amount of health. A move such as False Swipe will work for this. Here, the Pokémon has a higher chance of calling for help. It won't call for help if it currently has assistance.
Another way is to use the brand new item, the Adrenaline Orb. This item is found in all Pokémon Centers and, when used during a wild battle, will make the likelihood of the Pokémon calling for help even higher. You can only use one per battle, but the effect should last for the duration [of the battle]
Due to Adrenaline Orbs being usable only once per battle, as noted by @UnknownZombie in the comments, using additional Adrenaline Orbs is a very effective way to pass turns during battles where you are trying to chain SOS Pokémon, as you will not be able to use another Adrenaline Orb, but it will use up your turn.
A third way to improve your chaining probability is to use a Pokémon with the ability Intimidate, Unnerve, or Pressure.
The ability of the Pokémon you have in battle is also one that factors into S.O.S. Battles. If your Pokémon has the abilties Intimidate, Pressure or Unnerve then the opposing Pokémon is more likely to call for help.
Thus, the three main strategies are to keep the Pokémon at low health, to use an Adrenaline Orb, and to use a Pokémon with Intimidate, Pressure, or Unnerve.
As an aside:
Do note that even if a Pokémon calls for help, it is not guaranteed that a wild Pokémon will come to its aid.
Best Answer
The only way to turn away a Pokémon is to never tap them when they're visiting Isle Abeens and they still have a question mark over their head. If you've already tapped the Pokémon while it was visiting, it will not leave on its own and will have to be recruited and then released. This is not a problem since tapping a Pokémon doesn't provide any additional information.