Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)

We are very far from proving the existence of such an integer, let alone find an explicit value which works.

My question is:

What is the best known lower bound for $n$?

One way to obtain a lower bound $m$ for $n$ is to prove the existence of a curve of genus $2$ over $\mathbf{Q}$ with at least $m$ rational points.