If $f(x)$ is a linear function with $f(k)=4, f(f(k))=7$, and $f(f(f(k)))=19$, what is the value of $k$ ?
Since $f(k)=4$ and $f(f(k))=7$, then $f(4)=7$.

Since $f(f(k))=7$ and $f(f(f(k)))=19$, then $f(7)=19$.

Suppose that $f(x)=a x+b$ for some real numbers $a$ and $b$.

Since $f(4)=7$, then $7=4 a+b$.

Since $f(7)=19$, then $19=7 a+b$.

Subtracting, we obtain $(7 a+b)-(4 a+b)=19-7$ and so $3 a=12$ or $a=4$, which gives $b=7-4 a=7-16=-9$.

Therefore, $f(x)=4 x-9$ for all $x$.

Since $f(k)=4$, then $4=4 k-9$ and so $4 k=13$ or $k=\frac{13}{4}$.

ANSWER: $\frac{13}{4}$