6. As shown in Figure $8, \odot O$ has a radius of $1, B$ is any point on the circumference. The vertex $A$ of $\triangle A B C$ is also on the circumference, such that $A B = B C, \angle A B C=90^{\circ}$. What is the maximum length of $O C$?
6. $\sqrt{2}+1$.

By Ptolemy's theorem, we have
$$
A B \cdot C O \leqslant A O \cdot B C + B O \cdot A C,
$$

with equality if and only if points $A, B, C, O$ are concyclic.
Since $A B = B C$, it follows that $A C = \sqrt{2} A B$.
Therefore, $C O \leqslant 1 + \sqrt{2}$.