Is it doable for a player that play a reaction to add another reaction before the opponent ? Or by extension as much as reaction as he wants before the other ?

Yes — the player who plays a Reaction keeps priority and can play another Reaction (and another) before ever passing to the opponent. Priority only moves when you explicitly pass it (Rules 311–312, 338.1.b), so there's no automatic hand-off after one Reaction. You're limited only by what you can afford, since costs are paid as each card goes on the chain.

If you do pass and your opponent also passes, the newest item on the chain resolves — and you don't get another window to react to that same item. After it resolves, priority opens up again and you can still respond to whatever is left underneath. Note the flip side: if your opponent passes back to you while a chain is open, you can also just keep adding Reactions rather than letting things resolve — Actions are the ones that require an empty chain.