This question seems very basic but I cannot seem to find any literature on it.

Let $\mathbb{P}$ be a forcing notion. If $p$ is a condition of $\mathbb{P}$, define the predecessor set of $p$ to be $$\{q \in \mathbb{P} : p \leq_{\mathbb{P}} q \}.$$ Now assume every condition of $\mathbb{P}$ has a finite predecessor set. Is it possible for $\mathbb{P}$ to add no reals? If so, what is an example of such $\mathbb{P}$?