6. 10 .

Let the starting point of these vectors be the origin, and the endpoints of vectors $\boldsymbol{a}_{i} 、 \boldsymbol{b}_{j}$ be denoted as $A_{i} 、 B_{j}$. According to the geometric meaning of vector subtraction, point $B_{j}$ lies on the common points of circles with $A_{1} 、 A_{2}$ as centers and radii of $1,2,3$ respectively. It is not hard to find that there are 10 such points. Therefore, the maximum value of $k$ is 10.
6. Given $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in a plane, no two of which are parallel, satisfying $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\}(i=1,2 ; j=1,2, \cdots, k)$. Then the maximum value of $k$ is $\qquad$