Must all functionals be integrals and all path integrals over functions of functionals?

Question

I have two related questions:

1. Can you have a functional, $s[x(t)]$, which is not an integral? (I think a function of a functional is still a functional) Can you have a functional that has $x(t)$ not within an integral (or placed within an integral)?
2. Can you have a (Feynman) path integral $\int \mathscr{D}[x(\tau)] $ over a function which is not a function of a functional?

Reason Why I ask

The standard way of explaining path integrals is to split the path into $N$ segments and then take the limit as $N\rightarrow \infty$. That means we must be able to write: $$\lim_{N\rightarrow \infty}\int dx_1...dx_N f(x_1,...,x_N)$$ my confusion arises of what happens to $f(x_1,...,x_N)$ as $N\rightarrow \infty$. It must depend on the curve $x(\tau)$ but I am unsure of what forms this dependency takes.