One way to think about the fact that no surface in space can be triply ruled
Firstly, one sees that a surface $S$ of degree $3$ or higher is not ruled at all; a cubic surface can contain at most $27$ lines in total, and higher degree surfaces typically contain no lines at all. (One way to prove this is via an argument with incidence varieties; see the sketch in this tricki entry.)
So a non-planar ruled surface has to be a quadric (i.e. cut out by a degree $2$ equation).
On the other hand, if $\ell$ is a line passing lying on $S$, passing through a point $s \in S$, then $\ell$ lies in the tangent plane to $S$ at $s$. Since $S$ is a quadric, when you intersect it with a plane, the intersection is a (possibly degenerate) conic section, and so can contain at most two lines. Thus there are at most two lines on $S$ passing through any given point $s$, and hence a quadric is at most doubly ruled.