6. D.

First, regardless of the positional relationship between $a$, $b$, and $c$, we can always construct a parallelepiped $A B C D$ $A_{1} B_{1} C_{1} D_{1}$ such that $A B$ lies on line $a$, $B_{1} C_{1}$ lies on line $b$, and $D_{1} D$ lies on line $c$. Then, take any point $M$ on the extension of $D D_{1}$. Draw a plane $\alpha$ through $M$ and line $a$. Plane $\alpha$ intersects line $B_{1} C_{1}$ at point $P$ and line $A_{1} D_{1}$ at point $Q$, so $P Q \parallel a$. Therefore, in plane $\alpha$, line $P M$ is not parallel to $a$, and $P M$ must intersect $a$ at a point $N$. Thus, line $M N$ intersects lines $a$, $b$, and $c$ simultaneously. Since there are infinitely many ways to choose point $M$, there are infinitely many lines that intersect $a$, $b$, and $c$ simultaneously.