Up to 3 hits

Using Pets, you can possibly make a third hit if your pet has an AoE knockback summon. If the enemy is knocked back on heavy stander, a possible third hit may occur after the second hit activates said stander.

As far as normal usage, It's hard to normally make 3 hits. As someone who has been using puppetry since Day 1, I can assure at one point it did hit 3 times, and it was rank F when it happened. This was done at point blank range/normal melee distance. There was an unknown nerf that literally changed the timing between hits. I don't know all the details of 100% third hit, nor do I know about patch notes.