Three, and they have to alternate
The rule in the book: a team is entitled to a maximum of three hits, alternating between players. Both halves matter and the second half is the one people break.
You cannot receive a ball and then set it for yourself. Touch, partner, touch. That is the pattern, and it is why roundnet feels like volleyball for about four seconds before it stops feeling like volleyball at all.
You are not obliged to use all three. One touch straight back onto the net is legal, and against a team standing in the wrong place it is sometimes the best shot on the table. It is just rarely the percentage play.
What counts as one touch and what counts as two
This is where nearly every argument at a park comes from, and the rules are more generous than people assume.
On the first touch of a possession, the ball may hit various parts of the body at the same time and it still counts as one hit. Two hands, forearm and chest, hand and shoulder. If it is simultaneous, it is one.
Also on the first touch, the ball may contact several parts of one or both players' bodies one after another, as long as it all happens during a single action. Palms to fingers on the same hand is explicitly allowed, provided it does not become a catch or a throw. Still one hit.
The test the rulebook uses is whether the players moved in reaction to the original ball or to the new trajectory the first touch created. React to the original ball, it is one action.
The rule that lets you hit twice in a row
There is one, and beginners never know about it. After the first touch of the possession, a player may hit the ball a second time consecutively if their first hit happened before the ball began to travel downward. That counts as two hits, not one.
Two limits on it, both of them strict. It is not allowed on serve receive. And it cannot end in a hit on the net, so the second contact has to be a set to your partner.
In practice this is the rule that saves you when a set goes badly wrong and you have to rescue your own bad touch. Learn it and you stop conceding points you did not need to concede.
Two things that are always a violation
Catching. The ball must not come to rest on any part of you, and a catch is defined exactly that way: the ball coming to rest. This is the hardest thing to give up if you have played any sport where holding the ball is normal.
Two-handed hits. A player may not hit the ball with both hands, except on the first touch of a possession where simultaneous contacts are allowed. Outside that, one hand.
And the obvious one: four hits before returning to the net is a violation, and the point goes to the other team.
The one that catches out the receiving team
There is a rule that looks strange until you see the reason for it. If the receiving team returns the serve without using three touches, they lose the point.
So on serve receive you must use all three. You cannot receive it and knock it straight back. It exists to stop the receiving team from turning the serve into an instant counterattack, and it is one of the reasons rallies in this sport last as long as they do.
That is a rule the garden version usually ignores and every league in Europe does not.